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<pre><font size=+2>

<a name="HanLing">&lt;a name="HanLing"&gt;</a>
2011-07-09-14-38 start
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<a name="docA_001">&lt;a name="docA_001"&gt;</a>
<script language="javascript">link1()</script>
2011-07-05-19-10 始  a007051910
This file sphtrian.htm has a
function eyeUpMat(arg1,arg2,arg3)
Following explain how it works.
Input arguments are arg1,arg2,arg3
eyeUpMat() read two vectors arg1,
arg2 and property of vectors arg3
eyeUpMat() build a rotation matrix
from three input data.

<a name="docA_002">&lt;a name="docA_002"&gt;</a>
arg1 define a vector, which is a 
     coordinate axis vector for 
     the rotated system. 
arg1 define an accurate direction
arg1 vector components is measured
     from ground coordinate system.
arg2 define a vector, which is in 
     a coordinate plane of the 
     rotated system. 
arg2 define an approximate direction
arg2 vector components is measured
     from ground coordinate system.
arg3 regulate arg1,arg2 to be which
     coordinate in rotated system.
eyeUpMat() output a rotation matrix
from user given data.

<a name="docA_003">&lt;a name="docA_003"&gt;</a>
Avoid vector length to be a factor,
Require vector a (arg1) length=1
Require vector b (arg2) length=1
<!--2011-06-18-15-46 working time stamp-->
Require vector a be accurate.
Require vector b be approximate.
therefore, vector
  a=[a1,a2,a3] ---eqn.sph001
is one column in rotation matrix
[a1,a2,a3] is column one? or two?
or three? this is determined by 
arg3.

<a name="docA_004">&lt;a name="docA_004"&gt;</a>
Vector b is approximate direction
  b=[b1,b2,b3] ---eqn.sph002
do not show up in rotation matrix.

Assume vector a cross vector b get 
vector f
  f=[f1,f2,f3] ---eqn.sph003

<a name="docA_005">&lt;a name="docA_005"&gt;</a>
f=[f1,f2,f3]
 =[a1,a2,a3]×[b1,b2,b3]
 = ---eqn.sph004
[  i   j   k ]
[ a1  a2  a3 ]
[ b1  b2  b3 ]
 =
[a2b3-a3b2,
 a3b1-a1b3,
 a1b2-a2b1]
=[f1,f2,f3]

<a name="docA_006">&lt;a name="docA_006"&gt;</a>
<script language="javascript">link1()</script>
f1=a2b3-a3b2 ---eqn.sph005
f2=a3b1-a1b3 ---eqn.sph006
f3=a1b2-a2b1 ---eqn.sph007

<a name="docA_007">&lt;a name="docA_007"&gt;</a>
Vector a and vector b not required
to be perpendicular to each other.
Human brain estimation is hard to 
be exact perpendicular.

Vector f perpendicular to vector a.
Vector f perpendicular to vector b.
Above two are perpendicular exactly
because vector is created by math
calculation.
  [f1,f2,f3]=   ---eqn.sph004
  [a1,a2,a3]×[b1,b2,b3]

<a name="docA_008">&lt;a name="docA_008"&gt;</a>
But! vector f length is not one,
because vector a and vector b are
not perpendicular. Normalize vector 
f, let result be vector g
  g=[g1,g2,g3]  ---eqn.sph008
   =[f1,f2,f3]/|f|
In which 
  |f|=    ---eqn.sph009
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))

<a name="docA_009">&lt;a name="docA_009"&gt;</a>
Express g with given values 
[a1,a2,a3] and [b1,b2,b3], get
  g1=(a2b3-a3b2)/    ---eqn.sph010
     sqrt((a2b3-a3b2)*(a2b3-a3b2)
         +(a3b1-a1b3)*(a3b1-a1b3)
         +(a1b2-a2b1)*(a1b2-a2b1))
  g2=(a3b1-a1b3)/    ---eqn.sph011
     sqrt((a2b3-a3b2)*(a2b3-a3b2)
         +(a3b1-a1b3)*(a3b1-a1b3)
         +(a1b2-a2b1)*(a1b2-a2b1))
  g3=(a1b2-a2b1)/    ---eqn.sph012
     sqrt((a2b3-a3b2)*(a2b3-a3b2)
         +(a3b1-a1b3)*(a3b1-a1b3)
         +(a1b2-a2b1)*(a1b2-a2b1))
Normalized vector g length is one.

<a name="docA_010">&lt;a name="docA_010"&gt;</a>
Because vector g ⊥ vector a 
Because vector g ⊥ vector b 
plus g length be one, therefore
[g1,g2,g3] is one column in rotation
matrix. It is column one? or two?
or three? this is determined by 
arg3.

Now vector a and vector g are 
perpendicular and both length 
be one. The following find the
result of vector a × vector g

<a name="docA_011">&lt;a name="docA_011"&gt;</a>
<script language="javascript">link1()</script>
h=[h1,h2,h3]
 =[a1,a2,a3] × [g1,g2,g3]
 =       ---eqn.sph013
[  i   j   k ]
[ a1  a2  a3 ]
[ g1  g2  g3 ]
 =
[a2g3-a3g2,
 a3g1-a1g3,
 a1g2-a2g1]
=[h1,h2,h3]

<a name="docA_012">&lt;a name="docA_012"&gt;</a>
  h1=a2g3-a3g2 ---eqn.sph014
  h2=a3g1-a1g3 ---eqn.sph015
  h3=a1g2-a2g1 ---eqn.sph016

Above is h=a×g ---eqn.sph013

Some case need vector a × vector g 
Some case need vector g × vector a 

<a name="docA_013">&lt;a name="docA_013"&gt;</a>
Following is m=g×a ---eqn.sph017

m=[m1,m2,m3]
 =[g1,g2,g3] × [a1,a2,a3]
 =        ---eqn.sph017
[  i   j   k ]
[ g1  g2  g3 ]
[ a1  a2  a3 ]
 =
[-a2g3+a3g2,
 -a3g1+a1g3,
 -a1g2+a2g1]
=[m1,m2,m3]

<a name="docA_014">&lt;a name="docA_014"&gt;</a>
  m1=-a2g3+a3g2 ---eqn.sph018
  m2=-a3g1+a1g3 ---eqn.sph019
  m3=-a1g2+a2g1 ---eqn.sph020
<!-- 2011-06-18-17-42 -->
Use given data [a1,a2,a3] and [b1,b2,b3]
to express m=[m1,m2,m3] as following
m1=-a2(a1b2-a2b1)/  ---eqn.sph021
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   +a3(a3b1-a1b3)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
   +(a3b1-a1b3)*(a3b1-a1b3)
   +(a1b2-a2b1)*(a1b2-a2b1))
m2=-a3(a2b3-a3b2)/ ---eqn.sph022
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   +a1(a1b2-a2b1)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
m3=-a1(a3b1-a1b3)/ ---eqn.sph023
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   +a2(a2b3-a3b2)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))

<a name="docA_015">&lt;a name="docA_015"&gt;</a>
Use given data [a1,a2,a3] and [b1,b2,b3]
to express h=[h1,h2,h3] as following
h1=a2(a1b2-a2b1)/ ---eqn.sph024
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   -a3(a3b1-a1b3)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
h2=a3(a2b3-a3b2)/ ---eqn.sph025
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   -a1(a1b2-a2b1)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
h3=a1(a3b1-a1b3)/ ---eqn.sph026
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))
   -a2(a2b3-a3b2)/
  sqrt((a2b3-a3b2)*(a2b3-a3b2)
      +(a3b1-a1b3)*(a3b1-a1b3)
      +(a1b2-a2b1)*(a1b2-a2b1))

<a name="docA_016">&lt;a name="docA_016"&gt;</a>
<script language="javascript">link1()</script>
Above dropped multiplication "*"
Below add multiplication "*",
Because to verify equation, must
include "*"
m1=-a2*(a1*b2-a2*b1)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))+a3*(a3*b1-a1*b3)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph027

m2=-a3*(a2*b3-a3*b2)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))+a1*(a1*b2-a2*b1)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph028

m3=-a1*(a3*b1-a1*b3)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))+a2*(a2*b3-a3*b2)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph029
<a name="docA_017">&lt;a name="docA_017"&gt;</a>
h1=+a2*(a1*b2-a2*b1)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))-a3*(a3*b1-a1*b3)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph030

h2=+a3*(a2*b3-a3*b2)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))-a1*(a1*b2-a2*b1)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph031

h3=+a1*(a3*b1-a1*b3)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))-a2*(a2*b3-a3*b2)/sqrt((a2*b3-a3*b2)*(a2*b3-a3*b2)+(a3*b1-a1*b3)*(a3*b1-a1*b3)+(a1*b2-a2*b1)*(a1*b2-a2*b1))
   ---eqn.sph032
<!--2011-06-18-17-50 here-->
[m1,m2,m3] or [h1,h2,h3] be one 
column in a rotation matrix.

[m1,m2,m3] or [h1,h2,h3] is 
column one? or two? or three? 
this is determined by arg3.

<a name="docA_018">&lt;a name="docA_018"&gt;</a>
positive x=y×z , y=z×x , z=x×y   ---eqn.sph033
negative x=-z×y, y=-x×z, z=-y×x  ---eqn.sph034

h=[h1,h2,h3]   ---eqn.sph013
 =[a1,a2,a3] × [g1,g2,g3] ; h=a×g

m=[m1,m2,m3]   ---eqn.sph017
 =[g1,g2,g3] × [a1,a2,a3] ; m=g×a

<a name="docA_019">&lt;a name="docA_019"&gt;</a>
<font color=red><b>The following determine column order</b></font>
<!--


2011-06-18-18-19 here
Found error and deleted error eqn.
2011-06-19-19-11 made correction
The following is correct.



-->
If arg3='zx' or arg3=6
Assign <font color=red><b>accurate a=[a1,a2,a3] be z axis</b></font>
Approximate b=[b1,b2,b3] define x axis.
(vector b is not x axis, b define x)
[m1,m2,m3] or [h1,h2,h3] 
first : be one column in matrix
second: replace vector b=[b1,b2,b3] 
third : be x axis in rotated system.

<a name="docA_020">&lt;a name="docA_020"&gt;</a>
vector g ⊥ vector a （z axis）
vector g ⊥ vector b （x axis）
therefore <font color=red><b>y axis = vector g</b></font> =[g1,g2,g3]

Determine rotated system x axis:<font color=red size=+3><b>
x=y×z</b></font> , y=z×x , z=x×y 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=+z×y 
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=<font color=red><b>+y×z</b></font> 
 =[g1,g2,g3] × [a1,a2,a3]
drop h，because x=-z×y sign opposite
 use m，because x=y×z sign consistent
<font color=red size=+3><b>assign m be x</b></font>
code line: rot1=eyeUpMat(ptA,ptB,'zx');
<font color=red size=+3><b>rot.Mat is [m col.,g col.,a col.]</b></font>
2011-06-19-19-23 here
=====

<a name="docA_021">&lt;a name="docA_021"&gt;</a>
<script language="javascript">link1()</script>
If arg3='zy' or arg3=5
Assign accurate a=[a1,a2,a3] be z axis.
Approximate b=[b1,b2,b3] define y axis
[m1,m2,m3] or [h1,h2,h3] be y axis
vector g ⊥ vector a （z axis）
vector g ⊥ vector b （y axis）
therefore x axis = vector g =[g1,g2,g3]

<a name="docA_022">&lt;a name="docA_022"&gt;</a>
Determine rotated system y axis:
x=y×z , <font color=red><b>y=z×x</b></font> , z=x×y 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=<font color=red><b>+z×x</b></font>
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=+x×z 
 =[g1,g2,g3] × [a1,a2,a3]
 use h, since y=z×x sign consistent
drop m, since y=-x×z sign opposite
code line: rot1=eyeUpMat(ptA,ptB,'zy');
rot.Mat is [g col., h col., a col.]
2011-06-19-19-50 here
=====

<a name="docA_023">&lt;a name="docA_023"&gt;</a>
If arg3='yx' or arg3=3
Assign accurate a=[a1,a2,a3] be y axis
Approximate b=[b1,b2,b3] define x axis
[m1,m2,m3] or [h1,h2,h3] be x axis
vector g ⊥ vector a （y axis）
vector g ⊥ vector b （x axis）
therefore z axis = vector g =[g1,g2,g3]

<a name="docA_024">&lt;a name="docA_024"&gt;</a>
Determine rotated system x axis:<font color=red><b>
x=y×z</b></font> , y=z×x , z=x×y 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=<font color=red><b>+y×z</b></font> 
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=+z×y 
 =[g1,g2,g3] × [a1,a2,a3]
 use h, since x=+y×z sign consistent
drop m, since x=-y×z sign opposite

code line: rot1=eyeUpMat(ptA,ptB,'yx');
rot.Mat is [h col., a col., g col.]
2011-06-19-19-54 here
=====

<a name="docA_025">&lt;a name="docA_025"&gt;</a>
If arg3='yz' or arg3=4
Assign accurate a=[a1,a2,a3] be y axis
Approximate b=[b1,b2,b3] define z axis
[m1,m2,m3] or [h1,h2,h3] be z axis
vector g ⊥ vector a （y axis）
vector g ⊥ vector b （z axis）
therefore x axis = vector g =[g1,g2,g3]

<a name="docA_026">&lt;a name="docA_026"&gt;</a>
<script language="javascript">link1()</script>
Determine rotated system z axis:
x=y×z , y=z×x , <font color=red><b>z=x×y</b></font> 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=+y×x 
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=<font color=red><b>+x×y</b></font> 
 =[g1,g2,g3] × [a1,a2,a3]
drop h，because z=-y×x sign opposite
 use m，because z=+x×y sign consistent
code line: rot1=eyeUpMat(ptA,ptB,'yz');
rot.Mat is [g col., a col., m col.]
2011-06-19-19-57 here
=====

<a name="docA_027">&lt;a name="docA_027"&gt;</a>
If arg3='xy' or arg3=2
Assign accurate a=[a1,a2,a3] be x axis
Approximate b=[b1,b2,b3] define y axis
[m1,m2,m3] or [h1,h2,h3] be y axis
vector g ⊥ vector a （x axis）
vector g ⊥ vector b （y axis）
therefore z axis = vector g =[g1,g2,g3]

<a name="docA_028">&lt;a name="docA_028"&gt;</a>
Determine rotated system y axis:
x=y×z , <font color=red><b>y=z×x</b></font> , z=x×y 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=+x×z 
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=<font color=red><b>+z×x</b></font> 
 =[g1,g2,g3] × [a1,a2,a3]
drop h，because y=-x×z sign opposite
 use m，because y=+z×x sign consistent
code line: rot1=eyeUpMat(ptA,ptB,'xy');
rot.Mat is [a col., m col., g col.]
2011-06-19-20-08 here
=====

<a name="docA_029">&lt;a name="docA_029"&gt;</a>
If arg3='xz' or arg3=1
Assign accurate a=[a1,a2,a3] be x axis
Approximate b=[b1,b2,b3] define z axis
[m1,m2,m3] or [h1,h2,h3] be z axis
vector g ⊥ vector a （x axis）
vector g ⊥ vector b （z axis）
therefore y axis = vector g =[g1,g2,g3]

<a name="docA_030">&lt;a name="docA_030"&gt;</a>
Determine rotated system z axis:
x=y×z , y=z×x , <font color=red><b>z=x×y</b></font> 
x=-z×y, y=-x×z, z=-y×x 
h=[h1,h2,h3]=<font color=red><b>+x×y</b></font> 
 =[a1,a2,a3] × [g1,g2,g3]
m=[m1,m2,m3]=+y×x 
 =[g1,g2,g3] × [a1,a2,a3]
 use h, since z=+x×y sign consistent
drop m, since z=-y×x sign opposite
code line: rot1=eyeUpMat(ptA,ptB,'xz');
rot.Mat is [a col., g col., h col.]
2011-06-19-20-10 here

2011-07-05-20-53 stop a007052053

<a name="docA_031">&lt;a name="docA_031"&gt;</a>
<script language="javascript">link1()</script>
2011-07-05-22-50 start
Above has six different rotation
matrix, list them below
zx rot.Mat is [m col., g col., a col.] ---eqn.sph035
zy rot.Mat is [g col., h col., a col.] ---eqn.sph036
yx rot.Mat is [h col., a col., g col.] ---eqn.sph037
yz rot.Mat is [g col., a col., m col.] ---eqn.sph038
xy rot.Mat is [a col., m col., g col.] ---eqn.sph039
xz rot.Mat is [a col., g col., h col.] ---eqn.sph040
Although six look different, but
view from another point, they are
all the SAME, that is
<a name="docA_032">&lt;a name="docA_032"&gt;</a>
<font color=red><b>Rot.Mat must be [x col., y col., z col.] ---eqn.sph041</b></font>
<font color=red>
Rotated coordinate system is also
called bird body coordinate system.
It is used in Liu,Hsinhan page only.
</font>
If measured from ground coordinate 
system, bird body coordinate x axis 
is  x=[x1,x2,x3] ---eqn.sph042
If measured from ground coordinate 
system, bird body coordinate y axis 
is  y=[y1,y2,y3] ---eqn.sph043
If measured from ground coordinate 
system, bird body coordinate z axis 
is  z=[z1,z2,z3] ---eqn.sph044

<a name="docA_033">&lt;a name="docA_033"&gt;</a>
Rotation matrix has the following 
form
 [x1  y1  z1]
 [x2  y2  z2] ---eqn.sph045
 [x3  y3  z3]
This definition, change a point 
measured 
in bird body coordinate [Bx,By,Bz] 
to ground coordinate [Gx,Gy,Gz]
That is
 [Gx] [x1  y1  z1] [Bx]
 [Gy]=[x2  y2  z2]*[By] ---eqn.sph046
 [Gz] [x3  y3  z3] [Bz]
<a name="docA_034">&lt;a name="docA_034"&gt;</a>
It is easy to verify, for example
measured in bird body system
Bp=[1,0,0] ---eqn.sph047
convert to ground use next equation
 [Gx] [x1  y1  z1] [1]
 [Gy]=[x2  y2  z2]*[0] ---eqn.sph048
 [Gz] [x3  y3  z3] [0]
that is
 [Gx] [x1*1 + y1*0 + z1*0]
 [Gy]=[x2*1 + y2*0 + z2*0] ---eqn.sph049
 [Gz] [x3*1 + y3*0 + z3*0]
<a name="docA_035">&lt;a name="docA_035"&gt;</a>
that is
 [Gx] [x1]
 [Gy]=[x2] ---eqn.sph050
 [Gz] [x3]
This is perfect right, since rotation
matrix column one is defined to be 
bird body [1,0,0] measured at ground
coordinate system.
If bird Bp=[3,0,0], answer increase
or decrease in proportion.

<a name="docA_036">&lt;a name="docA_036"&gt;</a>
<script language="javascript">link1()</script>
If Bp=[0,1,0]，or Bp=[0,4,0]，
If Bp=[0,0,1]，or Bp=[0,0,5]，
the analysis is the same. Then a
bird point Bp=[3,4,5] transform 
to ground coordinate system is 
easy to understand.

<a name="docA_037">&lt;a name="docA_037"&gt;</a>
Many application need to convert
from ground to bird. In this case
transpose the rotation matrix
 [x1  y1  z1]
 [x2  y2  z2] ---eqn.sph045
 [x3  y3  z3]
to next form
 [x1  x2  x3]
 [y1  y2  y3] ---eqn.sph051
 [z1  z2  z3]
then
 [Bx] [x1  x2  x3] [Gx]
 [By]=[y1  y2  y3]*[Gy] ---eqn.sph052
 [Bz] [z1  z2  z3] [Gz]
is correct equation.

<a name="docA_038">&lt;a name="docA_038"&gt;</a>
Why arrange
 x=[x1,x2,x3]
 y=[y1,y2,y3]
 z=[z1,z2,z3]
to
 [x1  y1  z1]
 [x2  y2  z2] ---eqn.sph045
 [x3  y3  z3]
the result is a rotation matrix?
Please see <a href=http://freeman2.com/rotate3d.htm#docA_002>rotate3d.htm</a> or <a href=rotate3d.htm#docA_002>local</a>.
2011-07-05-23-26 stop
</font></pre>


<br><hr><hr><br>

<a name="QADraw">&lt;"QADraw"&gt;</a>
<script language="javascript">link1()</script>

<br>
<font color=red size=+2>
Spherical triangle graph page
</font>
<br>
<font color=red size=+3>
Hsinhan wait for wife Liling
<br>
until last breath !!!!!
<br>
鑫漢永遠等候愛妻麗玲，<br>至死為止
</font>
<a href="#HanLing">
why put here? 
</a> 2011-07-09-14-30
<br>

<font color=red size=+2><b>Please use MSIE browser to read this file</b></font>
<br>
<a href="#docB_002">
Graph area size</a>, W: <!--9810121216-->
  <INPUT id=QAboxWidth0
   value="360" size=3
   onchange="javascript:
if(QAboxWidth0.value<100||isNaN(QAboxWidth0.value))
QAboxWidth0.value=100; 
else 
if(QAboxWidth0.value>800)
QAboxWidth0.value=800;"
  >
H:
  <INPUT id=QAboxHeight0
   value="360" size=3
   onchange="javascript:
if(QAboxHeight0.value<100||isNaN(QAboxHeight0.value))
QAboxHeight0.value=100; 
else 
if(QAboxHeight0.value>800)
QAboxHeight0.value=800;"
  >

<INPUT onclick='QAboxWidth0.value=QAboxHeight0.value=""'
 type=button value="dW"><!--9911011628-->

<INPUT onclick='QAboxXmin0.value=QAboxXmax0.value=""'
 type=button value="dX">

<INPUT onclick='QAboxYmin0.value=QAboxYmax0.value=""'
 type=button value="dY">

<INPUT onclick='
QAboxWidth0.value=
QAboxHeight0.value=
QAboxXmin0.value=
QAboxYmin0.value=
QAboxXmax0.value=
QAboxYmax0.value="";
'
 type=button value="dAll">

<INPUT onclick='
QAboxWidth0.value=
QAboxHeight0.value="360"
QAboxXmin0.value=
QAboxYmin0.value="-1";
QAboxXmax0.value=
QAboxYmax0.value="1";
'
 type=button value="use6"><!--a005171016-->


<br>

x min:
  <INPUT id=QAboxXmin0
   value="-1" size=3
  >
, x max:
  <INPUT id=QAboxXmax0
   value="1" size=3
  >
; y min:
  <INPUT id=QAboxYmin0
   value="-1" size=3
  >
, y max:
  <INPUT id=QAboxYmax0
   value="1" size=3
  >
;
<br>

<font color=red>
<span id=QAspanAlert01>
x,y value below should be within range above
</span>
</font>


  <input onclick=" //a005171015
QAboxYmax0.value=
QAboxXmax0.value=1;
QAboxYmin0.value=
QAboxXmin0.value=-1;
"
 type="button" value="-1to1" /> 

<br>
<a name="lmnStar"></a><!--a007051130-->
Sph.triangle abc, given a,b and angle a & b; c is unknown.
<br>
vector a≡l,b≡m,p=l×m have auxiliary vector l*,m*,p*.
<br>
l,m,p all given, see <a href=tute0055.htm#a055_132>
tute0055.htm</a> eqn.rod136 to 139.
<br>
Draw l*,m*,p* or not <a href="#docB_003">?</a>
<input
 type="checkbox"
 id="QAchkbxID72"
 unchecked
 onclick="javascript:if(QAchkbxID72.checked)
QAspanAlert72.innerHTML='yes draw '
else
QAspanAlert72.innerHTML='not draw '
"
 />
<span id=QAspanAlert72>
not draw
</span><!--a007051136-->
l*,m*,p* 

  <input onclick=" //9912121154
QAboxYmax0.value=
QAboxXmax0.value=3;
QAboxYmin0.value=
QAboxXmin0.value=-3;
"
 type="button" value="-3 to 3" /> 

<br>
<font color=red><b>
l*,m*,p* length may be&gt;1. Please increase x/y range.
</b></font>

<br>
If draw l*,m*,p* click "Draw triangle AB side & angle"
<br>
or click "reverse". If click "Draw triangle ABC sides"
<br>
program ignore l*,m*,p* and not draw them.
<br>

l*,m*,p* color=<!--a007051206-->
<input id=QAstarclr value=red
 size=6
 />
, width=
<input id=QAstarwth value=1
 size=3
 />
, solidot=
<input id=QAstarsdd value=dot
 size=3
 />

<br>


<a name="QAground"></a>
<font color=red><b>ground</b></font>
 (fixed) 3D triad 

<input onclick="
QAstarclr.value=
QAstarwth.value=
QAstarsdd.value=''
"
 type="button" value="del l*,m*,p*" /> 

<br>
 color=
<input id=QAxzclr value=silver
 size=6
 />
, width=
<input id=QAxzwth value=2
 size=3
 />
, solidot=
<input id=QAxzsdd value=solid
 size=3
 />

<input onclick="
QAxzclr.value=
QAxzwth.value=
QAxzsdd.value=''
"
 type="button" value="dG" /> 

<br>

recover default color/width/solidot click

<input onclick="
QAaclr.value='#ffaaaa';
QAbclr.value='#66aa66';
QAcclr.value='#aaaaff';
QAxzclr.value='silver';
QAabcclr.value='black';
QAawth.value=
QAbwth.value=
QAcwth.value=1;
QAxzwth.value=2;
QAabcwth.value=2;
QAxzsdd.value=
QAabcsdd.value=
QAasdd.value=
QAbsdd.value=
QAcsdd.value='solid';
QAaxsclr.value='red'
QAaxswth.value=3
QAaxssdd.value='solid'
"
 type="button" value="cws" /> 

<a href="#docB_042">cws</a>

<br>
<a href="#docB_009">
b,c color</a>=
<input id=QAaclr value=#ffaaaa
 size=6
 />
, width=
<input id=QAawth value=1
 size=3
 />
, solidot=
<input id=QAasdd value=solid
 size=3
 />

<input onclick="
QAaclr.value=
QAawth.value=
QAasdd.value=''
"
 type="button" value="da" /> 

<br>

c,a color=
<input id=QAbclr value=#66aa66
 size=6
 />
, width=
<input id=QAbwth value=1
 size=3
 />
, solidot=
<input id=QAbsdd value=solid
 size=3
 />

<input onclick="
QAbclr.value=
QAbwth.value=
QAbsdd.value=''
"
 type="button" value="db" /> 

<br>

a,b color=
<input id=QAcclr value=#aaaaff
 size=6
 />
, width=
<input id=QAcwth value=1
 size=3
 />
, solidot=
<input id=QAcsdd value=solid
 size=3
 />

<input onclick="
QAcclr.value=
QAcwth.value=
QAcsdd.value=''
"
 type="button" value="dc" /> 

<br>


a,b,c color=
<input id=QAabcclr value=black
 size=6
 />
, width=
<input id=QAabcwth value=2
 size=3
 />
, solidot=
<input id=QAabcsdd value=solid
 size=3
 />

<input onclick="
QAabcclr.value=
QAabcwth.value=
QAabcsdd.value=''
"
 type="button" value="d" /> 

<br>

triangle color=
<input id=QAaxsclr value=red
 size=6
 />
, width=
<input id=QAaxswth value=3
 size=3
 />
, solidot=
<input id=QAaxssdd value=solid
 size=3
 />

<input onclick="
QAaxsclr.value=
QAaxswth.value=
QAaxssdd.value=''
"
 type="button" value="d" /> 

<br>
<a href="#docB_010">
tangent color</a>= <!--a007061001-->
<input id=QAtanclr value=red
 size=6
 />
, width=
<input id=QAtanwth value=1
 size=3
 />
, solidot=
<input id=QAtansdd value=dot
 size=3
 />

<input onclick="
QAtanclr.value=
QAtanwth.value=
QAtansdd.value=''
"
 type="button" value="d" /> 

<br>

Draw two tangent at vertex A▲
 <input
 type="checkbox"
 id="QAchkbxID61"
 /> 
<!--a006221113-->

; B◆ <input
 type="checkbox"
 id="QAchkbxID62"
 /> 

; C● <input
 type="checkbox"
 id="QAchkbxID63"
 /> 

 &nbsp; 
<input onclick="QAchkbxID61.checked=
QAchkbxID62.checked=
QAchkbxID63.checked=0
"
 type="button" value="d" /> 

<br>


Axes cross at x= 
<input id=QAxatL value=0
 size=3
 /> 
, y= 
<input id=QAyatL value=0
 size=3
 />
, 

  <input onclick="QAxatL.value=QAyatL.value=2"
 type="button" value="2" />

  <input onclick="QAxatL.value=QAyatL.value=0"
 type="button" value="0" />

 ; 
 <input
 type="checkbox"
 id="QAchkbxID51"
 onclick="javascript:if(QAchkbxID51.checked)
QAspanAlert51.innerHTML='Use x,y axis '
else
QAspanAlert51.innerHTML='Del x,y axis '
"
 /> 

<span id=QAspanAlert51>

</span>


<br>


<a name="axisAnglBox"></a>
<a href="#docB_012">
Next five boxes</a> convert φ,θ,ρ to x,y,z use degree.

<br>
Three numbers are latitude,longitude,radius. 

<br>
Equator is latitude=0. If click south/north be ▣ polar

<br>
latitude place read as polar angle. North pole=0.

<br>
If fill in just two numbers, program set radius=1.

<br>
"I" is eye ; "U" is up ; "A" is vertex A

<br>


<!--2011-06-09-08-36-->
<input onclick=QAboxEyeRho.value=''
 type="button" value="d" /> 
<input id=QAboxEyeRho value="0,12.5,1"
 size=20
 />

<input onclick="javascript:
var rr=1;if(QAchkbxID53.checked)rr=2;
QAeyeview.value=phirho2xyz(QAboxEyeRho.value,rr);
" type="button" value="I" /> ;
<!--a006090900-->


<input onclick=QAboxEyeRho.value=''
 type="button" value="d" /> 
<input id=QAbox_UpRho value="0,0,1.2"
 size=20
 />

<input onclick="javascript:
var rr=1;if(QAchkbxID53.checked)rr=2;
QA_upview.value=phirho2xyz(QAbox_UpRho.value,rr);
" type="button" value="U" /> ;
<!--a006090914-->

<br>

<input onclick=QAboxEyeRho.value=''
 type="button" value="d" /> 
<input id=QAboxARho value="16,75,1.0"
 size=20
 />

<input onclick="javascript:
var rr=1;if(QAchkbxID53.checked)rr=2;
QAboxVecA.value=phirho2xyz(QAboxARho.value,rr);
" type="button" value="A" /> 
<!--a006141942-->

 , 

<input onclick=QAboxEyeRho.value=''
 type="button" value="d" /> 
<input id=QAboxBRho value="48,-18,1.0"
 size=20
 />

<input onclick="javascript:
var rr=1;if(QAchkbxID53.checked)rr=2;
QAboxVecB.value=phirho2xyz(QAboxBRho.value,rr);
" type="button" value="B" /> 


<br>

<input onclick=QAboxEyeRho.value=''
 type="button" value="d" /> 
<input id=QAboxCRho value="36,8,1.0"
 size=20
 />

<input onclick="javascript:
var rr=1;if(QAchkbxID53.checked)rr=2;
QAboxVecC.value=phirho2xyz(QAboxCRho.value,rr);
" type="button" value="C" /> 
<!--a006141949-->


 south/north be <input
 type="checkbox"
 id="QAchkbxID53"
 onclick="javascript:if(QAchkbxID53.checked)
QAspanAlert53.innerHTML='polar '
else
QAspanAlert53.innerHTML='latitude '
"
 /> 
<span id=QAspanAlert53>
latitude 
</span><!--a006171620-->


<br>
Program do not read above five boxes, read only box below

<br>

<a name="QAeyeUpbox"></a>
View triangle at <a href="#docB_013">center</a>:
<span id=QAspanAlert10>

</span><!--a006162145-->

<br>

<a href="#docB_050">
eye xyz</a> 
<input id=QAeyeview value="0.5, 0.2, 0.35"
 size=30
 />

<input onclick="QAeyeview.value=''"
 type="button" value="dI" /> 

<input onclick="
if(QAspanAlert10.innerHTML.length>0)
{ //a006171130
var ww=QAspanAlert10.innerHTML;
QAeyeview.value=ww.substring(ww.indexOf('﹕')+1,ww.length)
QAspanAlert10.innerHTML='';
}
"
 type="button" value="center" /> 



<br>

Up xyz 
<input id=QA_upview value="0, 0, 1"
 size=30
 /><!--a005171038-->


<input onclick="QA_upview.value=''"
 type="button" value="dU" /> 

<input onclick="QA_upview.value='0, 0, 1'
QAeyeview.value='0.5, 0.2, 0.35'"
 type="button" value="use8" /> 


<br>
Click R, Record data to box17, auto ▣light;
<br>
Click A, Apply box17 data;
<br>
Click D, Delete box17 data, auto □light;
<br>
Save box17 data to your file for future use.

<br>


<input onclick="
QAboxVecA.value=QAboxVecB.value=''
QAboxVecC.value=''
"
 type="button" value="dABC" /> 

<input onclick="
QAboxVecA.value='1.1,2.2,3.3';
QAboxVecB.value='2.4,3.5,2.1'
QAboxVecC.value='3.6,2.7,1.8'
"
 type="button" value="uABC" /> 
 ; 

<input onclick="
QAboxAngA.value=QAboxAngB.value=''
"
 type="button" value="dAB" /> 

<input onclick="
{QAboxAngA.value='36';QAboxAngB.value='60'}
"
 type="button" value="uAB" /> 

; <a href="#QAbox17">record</a> 
<INPUT onclick="javascript:
QAboxc17.value=recordf(0);
QAchkbxID71.checked=1;
"
 type=button value="R">

<INPUT onclick="javascript:
eval(QAboxc17.value)
"
 type=button value="A">

<INPUT onclick="javascript:
QAboxc17.value='';
QAchkbxID71.checked=0
"
 type=button value="D">

<input
 type="checkbox"
 id="QAchkbxID71"
onclick="
if(QAboxc17.value.length>0)
QAchkbxID71.checked=1
else
QAchkbxID71.checked=0
"
 /><a href="#docB_015">light</a> <!--a006241247-->

<br>

<a name="ABCinBox"></a>
<a href="#docB_018">
A,B,C</a> is spherical triangle three vertex vectors. 
<br>
Given A,B (but not C) and angle A,B fill in next four boxes
<br>
angle must be degree, program do not cut angle to half.
<br>

angle:<input id=QAboxAngA value=24 size=25 /> 
; angle:<input id=QAboxAngB value=36 size=25 />

<br>
Given A,B,C (but not angles), fill in next three boxes

<br>

A:
<input id=QAboxVecA value="1.1,2.2,3.3"
 size=25
 />

; B:
<input id=QAboxVecB value="2.4,3.5,2.1"
 size=25
 />

<br>

C: <input id=QAboxVecC value=3.6,2.7,1.8 size=25 />
A,B,C length will be normalized.

<br>
<a name="QAClick"></a>
QADraw 
<input onclick="javascript:
drawSphereTriangle( //a006160935
   QAboxVecA.value,QAboxVecB.value,
   QAboxVecC.value)
recordf(); //a006240830
" type="button" value="Draw triangle
 ABC sides" /> ; 


<input onclick="javascript:
drawSphereTriangle( //a006141629
   QAboxVecA.value,QAboxVecB.value
  ,QAboxAngA.value,QAboxAngB.value)
recordf(); //a006240830
" type="button" value="Draw triangle 
AB side & angle" /> 

<input onclick="javascript:drawSphereTriangle(QAboxVecA.value,QAboxVecB.value,QAboxAngA.value,QAboxAngB.value,1)" type="button" value="reverse" />
 ; 


<input onclick="drawSphereTriangle(0);
QAspanEqnDoc10.innerHTML=''
"
 type="button" value="wipe" /> 

<br>


<a name="QAgraphDiv"></a> 

<div id=QAdivGraph10></div>


<a name="graphDoc"></a> 
<span id=QAspanEqnDoc10> 
<br>
</span>

<br> 
<font color=red size=+2><b>Please use MSIE browser to read this file</b></font>
<br>

<a name="QAbox11"></a> 
  <input onclick=" //a007041909
runTestCodeQA(QAboxc11.value);
"
 type="button" value="%% draw Box11 code" /> 
<br>

Box11, output XYGraph code 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc11").value.toString())'
 type=button value="copy11">
<INPUT onclick='document.getElementById("QAboxc11").value=""'
 type=button value="del 11">
 

<br><!--a006141528 use QAboxc11-->
<TEXTAREA
 id=QAboxc11
 name=QAboxc11 rows=6
 cols=65
 >
</TEXTAREA>

<br>
<a name="QAbox12"></a> 

Box12, output 2D x,y data 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc12").value.toString())'
 type=button value="copy12">
<INPUT onclick='document.getElementById("QAboxc12").value=""'
 type=button value="del 12">
;

<br><!--a006141530 use QAboxc12-->
<TEXTAREA
 id=QAboxc12
 name=QAboxc12 rows=6
 cols=65
 >
</TEXTAREA>

<br>

<a name="QAbox13"></a> 

Box13, output 3D x,y,z data 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc13").value.toString())'
 type=button value="copy13">
<INPUT onclick='document.getElementById("QAboxc13").value=""'
 type=button value="del 13">
;

<br><!--a006141655 use QAboxc13-->
<TEXTAREA
 id=QAboxc13
 name=QAboxc13 rows=6
 cols=65
 >
</TEXTAREA>

<br>

<a name="QAbox14"></a> 

Box14, vector,angle,area,<a href="#docB_025">normal</a> 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc14").value.toString())'
 type=button value="copy14">
<INPUT onclick='document.getElementById("QAboxc14").value=""'
 type=button value="del 14">
;

<br><!--a006142133 use QAboxc14-->
<TEXTAREA
 id=QAboxc14
 name=QAboxc14 rows=6
 cols=65
 >
</TEXTAREA>

<br>
<a name="QAbox15"></a> 

Box15, rotation matrix 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc15").value.toString())'
 type=button value="copy15">
<INPUT onclick='document.getElementById("QAboxc15").value=""'
 type=button value="del 15">
;

<br><!--a006162139 use QAboxc15-->
<TEXTAREA
 id=QAboxc15
 name=QAboxc15 rows=6
 cols=65
 >
</TEXTAREA>

<br>

<input onclick="javascript:
QAboxc16.value=''
+'[EyeUp Mat] test function eyeUpMat(pA,pB,cx). Input from\n'
+'Box15. Need two vectors. One line one vector, one line \n'
+'three numbers. Click 31,32,33 to get examples. pA,pB are \n'
+'from Box15. pA is accurate direction, pB is approximate \n'
+'direction. cx is type. cx=1,2,3,4,5,6。 function eyeUpMat()\n'
+'read input and return correct rotation matrix. In 3D space,\n'
+' pA has three choices, x,y,z and pB has two choices 3*2=6 \n'
+'choices over all.\n'

var pA,pB,L,rotM,qA,qB,ans1='';
var in0=QAboxc15.value
if(in0.length>0){
if(in0.indexOf('\n')>0){
in0=in0.split('\n');
in0[0]=cleanData(in0[0])
pA=in0[0].replace(/ /g,',')
in0[1]=cleanData(in0[1])
pB=in0[1].replace(/ /g,',')

rotN=eyeUpMat(pA,pB,1);
if(typeof(rotN)=='object'){ //a007042221 if()
rotM=transpoMat(rotN);
pA=pA.split(',');pB=pB.split(',');
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,1)=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='xz qA='+qA+'\n'+'1; qB='+qB+'\n'
ans1+='###above is xz, x accurate , z approximate.\n'
     +'###after rotation pA become qA=[x,0,0]\n'
ans1+='###pB become qB=[x,0,z] after rotation qB stay\n'
     +'###on xz plane, thefore qB y coordinate=0\n'
ans1+='Next change third parameter.\n';}

rotN=eyeUpMat(pA,pB,2);
if(typeof(rotN)=='object'){ 
rotM=transpoMat(rotN);
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,2)=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='xy qA='+qA+'\n'+'2; qB='+qB+'\n'
ans1+='###above is xy, x accurate , y approximate. \n'
     +'###after rotation pA become qA=[x,0,0]\n'
ans1+='###pB become qB=[x,y,0] after rotation qB stay\n'
     +'###on xy plane, thefore qB z coordinate=0\n'
ans1+='Next change third parameter.\n';}
else ans1+=rotN+'\n';



rotN=eyeUpMat(pA,pB,3);
if(typeof(rotN)=='object'){ 
rotM=transpoMat(rotN);
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,3)=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='yx qA='+qA+'\n'+'3; qB='+qB+'\n'
ans1+='###above is yx, y accurate , x approximate.\n'
     +'###after rotation pA become qA=[0,y,0]\n'
ans1+='###pB become qB=[x,y,0] after rotation qB stay\n'
     +'###on xy plane, thefore qB z coordinate=0\n'
ans1+='Next change third parameter.\n';}


rotN=eyeUpMat(pA,pB,4);
if(typeof(rotN)=='object'){ 
rotM=transpoMat(rotN);
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,4)=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='yz qA='+qA+'\n'+'4; qB='+qB+'\n'
ans1+='###above is yz, y accurate , z approximate.\n'
     +'###after rotation pA become qA=[0,y,0]\n'
ans1+='###pB become qB=[0,y,z] after rotation qB stay\n'
     +'###on yz plane, thefore qB x coordinate=0\n'
ans1+='Next change third parameter.\n';}

rotN=eyeUpMat(pA,pB,5);
if(typeof(rotN)=='object'){ 
rotM=transpoMat(rotN);
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,5)=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='zy qA='+qA+'\n'+'5; qB='+qB+'\n'
ans1+='###above is zy, z accurate , y approximate.\n'
     +'###after rotation pA become qA=[0,0,z]\n'
ans1+='###pB become qB=[0,y,z] after rotation qB stay\n'
     +'###on yz plane, thefore qB x coordinate=0\n'
ans1+='Next change third parameter.\n';}

rotN=eyeUpMat(pA,pB,'zx');
if(typeof(rotN)=='object'){ 
rotM=transpoMat(rotN);
qA=mAvB(rotM,pA);qB=mAvB(rotM,pB);
ans1+='eyeUpMat(pA,pB,\'zx\')=rotN=\n'+mat2str(rotN)
ans1+='transpoMat(rotN)=rotM=\n'+mat2str(rotM)
ans1+='zx qA='+qA+'\n'+'6; qB='+qB+'\n'
ans1+='###above is zx, z accurate , x approximate.\n'
     +'###after rotation pA become qA=[0,0,z]\n'
ans1+='###pB become qB=[x,0,z] after rotation qB stay\n'
     +'###on zx plane, thefore qB y coordinate=0\n';}

QAboxc16.value+=''
+'Input pA='+pA+'\n'
+'point A location measured from ground coordinate.\n'
+'Input pB='+pB+'\n'
+'point B location measured from ground coordinate.\n'
+'Output qA and qB This is point A and point B measured\n'
+'from rotated coordinate. Point A and point B stay in \n'
+'space not moving. Two different coordinate system get\n'
+'two different location measure.\n'
+'rotM=transpoMat(rotN); rotN is rotation matrix, \n'
+'rotM is transpose of rotN.\n'
+'qA and qB=rotM multiply by pA and pB;\n'
+'pA and pB=rotN multiply by qA and qB.\n'
+'qA and qB below is rotM multiply by pA and pB. \n'
+'Code is qA=mAvB(rotM,pA)\n'
+ans1+'a006192056dbg\n'
+'function eyeUpMat() accept the following replacement.\n'
+'[xz] can be replaced by 1 ，[xy] can be replaced by 2\n'
+'[yx] can be replaced by 3 ，[yz] can be replaced by 4\n'
+'[zy] can be replaced by 5 ，[zx] can be replaced by 6\n'
+'a006192130 test eyeUpMat(pA,pB,6) result look ok.\n'
}else{} //two else{} hint move to top. When Box15 has
}else{} //text (error case) still provide explanation. a007041715
" type="button" value="EyeUp Mat" /> 


<input onclick="javascript:
QAboxc15.value='1.1, 2.2, 3.3\n2.4,3.5,2.1\n';
" type="button" value="31" /> 
<input onclick="javascript:
QAboxc15.value='2.4,3.5,2.1\n3.6,2.7,1.8\n';
" type="button" value="32" /> 
<input onclick="javascript:
QAboxc15.value='3.6,2.7,1.8\n1.1, 2.2, 3.3\n';
" type="button" value="33" /> 
<!--a006241700-->

; 

<input onclick="javascript:
QAboxc16.value=transpoMat(QAboxc15.value)
" type="button" value="transpose" /> 
<!--a006190003-->


<input onclick="javascript:
QAboxc15.value='sin(a),cos(b)\nexp(c),log(d)\ntan(e),cot(f)\n';
" type="button" value="21" /> 
<!--a006190541-->

<input onclick="javascript:
QAboxc15.value='1.1, 2.2, 3.3\n4.4, 5.5, 6.6\n';
" type="button" value="22" /> 
<!--a006191209-->

; 

<input onclick="javascript:
QAboxc15.value='1.1,2.2,6\n2.4,7,2.1\n8,2.7,1.8\n';
QAeyeview.value='0.506,0.61,0.539'
QA_upview.value='0, 0, 1'
QAchkbxID61.checked=
QAchkbxID62.checked=
QAchkbxID63.checked=false
" type="button" value="34" /> 
<!--a006241756-->

<!--a006251132 ABC box -->
to ABC <a href="#ABCinBox">box </a>

<br>


<!--a007041939-->
<input onclick="javascript:
QAboxc15.value=QAboxVecA.value+'\n'
+QAboxVecB.value+'\n'+QAboxVecC.value
" type="button" value="read" /> 


<!--a006241758-->
<input onclick="javascript:
var in0=QAboxc15.value;
if(in0.indexOf('\n')<0)
QAboxc17.value='Box15 is empty. '
+polarTriHint;
else{
in0=in0.split('\n');
in0[0]=cleanData(in0[0])
pA=in0[0].split(' ')
in0[1]=cleanData(in0[1])
pB=in0[1].split(' ')
in0[2]=cleanData(in0[2])
pC=in0[2].split(' ')

QAboxc16.value=''
+vec1(v1Xv2(pC,pA))+'\n'
+vec1(v1Xv2(pB,pC))+'\n'
+vec1(v1Xv2(pA,pB))+'\n' }
//Above order allow dual of dual
//return to original. If change
//order, dual of dual reverse。
//a006241840
" type="button" value="dual triangle" /> 


<INPUT name="QARadio1" type="radio"
>1 
<INPUT name="QARadio1" type="radio"
>2 
<INPUT name="QARadio1" type="radio"
>3 
<INPUT name="QARadio1" type="radio"
>4 


<input onclick="javascript:
for(var i=0;i<QARadio1.length;i++)
if(QARadio1[i].checked)break; i++;
fillBoxABC(QAboxc16.value,i)
" type="button" value="fill" />
<!--a006251129-->

<INPUT name="QARadio1" type="radio"
>5 
<INPUT name="QARadio1" type="radio"
>6 
<INPUT name="QARadio1" type="radio"
>7 
<INPUT name="QARadio1" type="radio"
 checked
>8 

<br>
<a name="QAbox16"></a> 

Box16, debug 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc16").value.toString())'
 type=button value="copy16">
<INPUT onclick='document.getElementById("QAboxc16").value=""'
 type=button value="del 16">
;

<input onclick="javascript:
QAboxc15.value=QAboxc16.value
QAboxc16.value=''
" type="button" value="16 to 15" /> 
<!--a006241828-->

<a href="#docB_026">EyeUp/Dual</a>

<br>

<script language="javascript">link1()</script>

<br><!--a006190537 use QAboxc16-->
<TEXTAREA
 id=QAboxc16
 name=QAboxc16 rows=6
 cols=65
 >
</TEXTAREA>

<br>
<a name="QAbox17"></a> 


<input onclick="javascript:
QAboxc16.value=''
+'0.1696327431206737,0.3392654862413474,0.9252695079309474\n'
+'0.3120042120852949,0.9100122852487769,0.27300368557463306\n'
+'0.9266685553992929,0.31275063744726133,0.20850042496484092\n'
+'-0.24570116952620294,0.9237882207186159,-0.2936771331836886\n'
+'-0.13446793428893575,-0.2421587041740142,0.9608733197674024\n'
+'0.9496708166757841,-0.30715677296977606,-0.061482166301642556\n'
+'First three lines are a,b,c vectors original triangle vertex.\n'
+'Second three lines are x,y,z vectors dual triangle vertex.\n'
+'Click [verify01] button find a dot which two of x,y,z get 0.\n'
+'find b perpendicular to which two of x,y,z Similarly,\n'
+'find c ⊥ to which two. QCDraw fig.6 need this function.\n'
+'Input at Box16, output to Box17 too.\n'
" type="button" value="35" /> 


<input onclick="javascript:
//向量 A 點積向量 B
function AdB(A,B){var ans=(A[0]*B[0]+A[1]*B[1]+A[2]*B[2]);
if(Math.abs(ans)<1.e-8)return 0; return ans; }

/**分離，變為讀取資料（萬用），不要鎖在程式內（無用）
var a=[0.1696327431206737,0.3392654862413474,0.9252695079309474]
var b=[0.3120042120852949,0.9100122852487769,0.27300368557463306]
var c=[0.9266685553992929,0.31275063744726133,0.20850042496484092]
var x=[-0.24570116952620294,0.9237882207186159,-0.2936771331836886]
var y=[-0.13446793428893575,-0.2421587041740142,0.9608733197674024]
var z=[0.9496708166757841,-0.30715677296977606,-0.061482166301642556]
/**/
var in0=QAboxc16.value;
if(in0.indexOf('\n')>0){
in0=in0.split('\n');
for(var i=0;i<6;i++){
in0[i]=in0[i].split(',');
for(var j=0;j<3;j++){
in0[i][j]=parseFloat(in0[i][j]);}}
var a,b,c,x,y,z;
a=in0[0];b=in0[1];c=in0[2];
x=in0[3];y=in0[4];z=in0[5];
var axy='axy='+AdB(a,x)+','+AdB(a,y)+'\n';
var axz='axz='+AdB(a,x)+','+AdB(a,z)+'\n';
var ayz='ayz='+AdB(a,y)+','+AdB(a,z)+'\n';
var bxy='bxy='+AdB(b,x)+','+AdB(b,y)+'\n';
var bxz='bxz='+AdB(b,x)+','+AdB(b,z)+'\n';
var byz='byz='+AdB(b,y)+','+AdB(b,z)+'\n';
var cxy='cxy='+AdB(c,x)+','+AdB(c,y)+'\n';
var cxz='cxz='+AdB(c,x)+','+AdB(c,z)+'\n';
var cyz='cyz='+AdB(c,y)+','+AdB(c,z)+'\n';
QAboxc17.value='' 
+'a='+a+'\n'
+'b='+b+'\n'
+'c='+c+'\n'
+'x='+x+'\n'
+'y='+y+'\n'
+'z='+z+'\n'
+'Two vectors dot result as following\n'
+axy+axz+ayz
+bxy+bxz+byz
+cxy+cxz+cyz
+'Three of above data are [0,0], those three \n'
+'sets can be used to draw semilunar triangle.\n'
+'If input come from [35], eye and screen_up are\n'
+'eye 0.506,0.61,0.539\n'
+' up 0, 0, 1\n'
+'a006251312'; }
else QAboxc17.value=
'Please fill data to Box16 or click [35] to find example.' 
" type="button" value="verify01" /> 

<a href="#docB_031">verify help</a>

 ; 

<input onclick="javascript:
QAboxc17.value=mat33det(QAboxc16.value)
" type="button" value="33MatDet" /> 
<!--a007091814-->
 ; 

<a href="#docB_016">record Help</a>

<a href="#QAClick">draw click</a>
<br>

Box17, record, apply 

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QAboxc17").value.toString())'
 type=button value="copy17">
<INPUT onclick='document.getElementById("QAboxc17").value=""
QAchkbxID71.checked=0;
'
 type=button value="del 17">
;
<INPUT onclick="javascript:
QAboxc17.value=recordf(0)
QAchkbxID71.checked=1;
"
 type=button value="record">

<INPUT onclick="javascript:eval(QAboxc17.value)"
 type=button value="apply">

<br><!--a006240746 use QAboxc17-->
<TEXTAREA
 id=QAboxc17
 name=QAboxc17 rows=6
 cols=65
 >
</TEXTAREA>

<br>

QAboxc17.value='string' 

<br>


<script lang="javascript">
<!--

polarTriHint=''
+'Please fill 3 vertex vectors to Box15.\n'
+'or click [34] for example, or click [read] to read\n'
+'A,B,C vectors from input box above QADrawing board.\n'
+'Data has three lines, each line three numbers.\n'
+'Next click [dual triangle] button. Program create\n'
+'three vectors which is dual triangle of the original.\n'
+'Output to box16. Select one of eight radio buttons,\n'
+'then click [fill] button. Program send three dual \n'
+'vertex values to A,B,C box above drawing board.\n'
+'Click [box] in [to ABC box]. next click [Draw triangle\n'
+'ABC sides] button. Drawing board display dual triangle.\n'
+'Each vertex has +/- two choices, three has 2*2*2=8\n'
+'choices. Only one is correct, other seven are wrong.\n'
+'Correct dual and original shape like [◎] one contain\n'
+'the other one. Wrong one no such relation. a006251215\n'
;

function fillBoxABC(arg1,arg2)
{ //a006251136

var in0=arg1;
var do8=arg2;
if(in0.indexOf('\n')<0){
QAboxc17.value='Box16 no data. '+polarTriHint;
;return;}
in0=in0.split('\n')
in0[0]=cleanData(in0[0])
in0[0]=in0[0].split(' ')
in0[1]=cleanData(in0[1])
in0[1]=in0[1].split(' ')
in0[2]=cleanData(in0[2])
in0[2]=in0[2].split(' ')
for(var i=0;i<3;i++)
for(var j=0;j<3;j++)
in0[i][j]=parseFloat(in0[i][j]);
if(do8==1) //x+,y+,z+
  {
QAboxVecA.value=+in0[0][0]+','
+(+in0[0][1])+','+(+in0[0][2]);
QAboxVecB.value=+in0[1][0]+','
+(+in0[1][1])+','+(+in0[1][2]);
QAboxVecC.value=+in0[2][0]+','
+(+in0[2][1])+','+(+in0[2][2]);
  }
else
if(do8==2) //x+,y+,z-
  {
QAboxVecA.value=+in0[0][0]+','
+(+in0[0][1])+','+(+in0[0][2]);
QAboxVecB.value=+in0[1][0]+','
+(+in0[1][1])+','+(+in0[1][2]);
QAboxVecC.value=-in0[2][0]+','
+(-in0[2][1])+','+(-in0[2][2]);
  }
else
if(do8==3) //x+,y-,z+
  {
QAboxVecA.value=+in0[0][0]+','
+(+in0[0][1])+','+(+in0[0][2]);
QAboxVecB.value=-in0[1][0]+','
+(-in0[1][1])+','+(-in0[1][2]);
QAboxVecC.value=+in0[2][0]+','
+(+in0[2][1])+','+(+in0[2][2]);
  }
else
if(do8==4) //x+,y-,z-
  {
QAboxVecA.value=+in0[0][0]+','
+(+in0[0][1])+','+(+in0[0][2]);
QAboxVecB.value=-in0[1][0]+','
+(-in0[1][1])+','+(-in0[1][2]);
QAboxVecC.value=-in0[2][0]+','
+(-in0[2][1])+','+(-in0[2][2]);
  }
else
if(do8==5) //x-,y+,z+
  {
QAboxVecA.value=-in0[0][0]+','
+(-in0[0][1])+','+(-in0[0][2]);
QAboxVecB.value=+in0[1][0]+','
+(+in0[1][1])+','+(+in0[1][2]);
QAboxVecC.value=+in0[2][0]+','
+(+in0[2][1])+','+(+in0[2][2]);
  }
else
if(do8==6) //x-,y+,z-
  {
QAboxVecA.value=-in0[0][0]+','
+(-in0[0][1])+','+(-in0[0][2]);
QAboxVecB.value=+in0[1][0]+','
+(+in0[1][1])+','+(+in0[1][2]);
QAboxVecC.value=-in0[2][0]+','
+(-in0[2][1])+','+(-in0[2][2]);
  }
else
if(do8==7) //x-,y-,z+
  {
QAboxVecA.value=-in0[0][0]+','
+(-in0[0][1])+','+(-in0[0][2]);
QAboxVecB.value=-in0[1][0]+','
+(-in0[1][1])+','+(-in0[1][2]);
QAboxVecC.value=+in0[2][0]+','
+(+in0[2][1])+','+(+in0[2][2]);
  }
else
if(do8==8) //x-,y-,z-
  {
QAboxVecA.value=-in0[0][0]+','
+(-in0[0][1])+','+(-in0[0][2]);
QAboxVecB.value=-in0[1][0]+','
+(-in0[1][1])+','+(-in0[1][2]);
QAboxVecC.value=-in0[2][0]+','
+(-in0[2][1])+','+(-in0[2][2]);
  }
//a006251150
} //function fillBoxABC(arg1,arg2)





//record 3D box data and
//output 3D box data 
//sphere3c.htm  sphere3d.htm
//function inOut3D(arg1) //9912171126
//
//rodriguc.htm  rodrigue.htm
//a006240820,a006240825 simple version
//a006241124,a006241221 better version
function recordf(arg1)
{ //a006241124
if(arg1==2) //evaluation, not record.
  {
eval(QAboxc17.value);
return; //done evaluation, return.
  }

var oStr=''; 
var aStr,bStr,cStr;

var q='QA';
var v;

var bx='box'; 

v='.value';

a=[];
a[0]='boxAng';
a[1]='boxVec';

b=[];
b[0]='A';
b[1]='B';
b[2]='C';

for(i=0;i<a.length;i++)
for(j=0;j<b.length;j++)
  { //a006241213
if(i==0&&j==2)continue;
bStr=q+a[i]+b[j]+v ;
oStr+=''+bStr+'="'+ eval(bStr) +'"\n'
  }

///////////

oStr+=''
+'QAeyeview.value="'+QAeyeview.value+'"\n'
+'QA_upview.value="'+QA_upview.value+'"\n'

///////////

var a=[];
a[0]='a';
a[1]='b';
a[2]='c';
a[3]='axs';
a[4]='abc';
a[5]='xz';
a[6]='star'; //a007051227

var b=[];
b[0]='clr';
b[1]='sdd';
b[2]='wth';

var i,j,k;
for(i=0;i<a.length;i++)
for(j=0;j<b.length;j++)
  { //
aStr=q+a[i]+b[j]+v 
oStr+=''+aStr+'="'+ eval(aStr) +'"\n'
  }

///////////

a='chkbxID6'
v='.checked'

for(i=0;i<3;i++)
  { 
bStr=q+a+(i+1)+v ;
oStr+=''+bStr+'='+ eval(bStr) +'\n'
  }

bStr=q+'chkbxID72'+v; //a007070004
oStr+=''+bStr+'='+ eval(bStr) +'\n'

///////////

v='0.value'; 

b=[];
b[0]='Width';
b[1]='Height';
b[2]='Xmin';
b[3]='Xmax';
b[4]='Ymin';
b[5]='Ymax';

for(i=0;i<b.length;i++)
  { //
bStr=q+bx+b[i]+v ;
oStr+=''+bStr+'="'+ eval(bStr) +'"\n'
  }

///////////


oStr=''
+'//Spherical triangle graph page created data. Paste\n'
+'//to Box17 of http://freeman2.com/sphtrian.htm \n'
+'//Click [apply] button rebuild input data. Next click\n'
+'//[Draw triangle ABC sides] button to draw graph.\n'
+'//Data created on '+yearMdhm()+'\n'
+oStr

return oStr;

} //function recordf(arg1) a006241221




//matrix A, vector B multiplication.
function mAvB(A,B){
return [A[0][0]*B[0]+A[0][1]*B[1]+A[0][2]*B[2],
A[1][0]*B[0]+A[1][1]*B[1]+A[1][2]*B[2],
A[2][0]*B[0]+A[2][1]*B[1]+A[2][2]*B[2]];
}









//經度	longitude
//緯度	latitude
//
function xyz2LatitudeLongitude(
 arg1 //point A x,y,z coordinate
,arg2 //arg2==1 output polar angle & longitude
  )  //arg2 NOT=1 output latitude & longitude
{ //a006181037

var i,j,k;
var L;

var ptA=arg1;
if(typeof(ptA)=='string')
  {
if(ptA.indexOf(',')>0)
ptA=ptA.split(',');
else{
while(ptA.indexOf('  ')>=0)
ptA.replace('  ',' ');
var i=0;var j=ptA.length-1;
if(ptA.charAt(i)==' ')i++;
if(ptA.charAt(j)==' ')j--;
ptA=ptA.substring(i,j+1);
ptA=ptA.split(' ');}
for(i=0;i<ptA.length;i++)
ptA[i]=parseFloat(ptA[i]);
  }
if(typeof(ptA)!='object'
 ||ptA.length!=3)
return 'xyz2LatitudeLongitude() error 01'

function L1(A){//a006181057 try function in function 
return Math.sqrt(A[0]*A[0]+A[1]*A[1]+A[2]*A[2]);}

L=L1(ptA);
if(Math.abs(L)<1.e-8)
return 'xyz2LatitudeLongitude() error 02'

for(i=0;i<ptA.length;i++)ptA[i]=ptA[i]/L;
var Latitud0,Lat0;
var Longitu0;
Lat0= //a006191754 Lat0
Latitud0=Math.asin(ptA[2])*180/Math.PI;
if(arguments.length>1&&arg2==1)
//Latitud0=Math.PI-Latitud0;
Latitud0=90-Latitud0; //a006191755
if(Math.abs(Lat0-90)<1.e-8)Longitu0='arbitrary value'
else //north pole and south pole both longitude arbitrary 
if(Math.abs(Lat0+90)<1.e-8)Longitu0='arbitrary value'
else
Longitu0=Math.atan(ptA[1]/ptA[0])*180/Math.PI;
//經度	longitude
//緯度	latitude
return [Latitud0,Longitu0]; //a006181201

} //function xyz2LatitudeLongitude()



//a006200826
//arg1[0]=N/S latitude 
//arg1[1]=E/W longitude
function LatitudeLongitude2xyz(
 arg1 //input point A latitude,longitude
,arg2 //arg2==1 input polar angle & longitude
  )   //arg2 NOT=1 input Latitude & Longitude
{ //output x,y,z
var i,j,k;
var x,y,z;

var ptA=arg1;
if(typeof(ptA)=='string')
  {
if(ptA.indexOf(',')>0)
ptA=ptA.split(',');
else{
while(ptA.indexOf('  ')>=0)
ptA.replace('  ',' ');
i=0; j=ptA.length-1;
if(ptA.charAt(i)==' ')i++;
if(ptA.charAt(j)==' ')j--;
ptA=ptA.substring(i,j+1);
ptA=ptA.split(' ');}
for(i=0;i<ptA.length;i++)
ptA[i]=parseFloat(ptA[i]);
  }
if(typeof(ptA)!='object'
 ||ptA.length!=2)
return 'LatitudeLongitude2xyz() error 01'

if(arguments.length>1&&arg2==1)
ptA[0]=90-ptA[0];

ptA[0]=ptA[0]*Math.PI/180; //N/S latitude 
ptA[1]=ptA[1]*Math.PI/180; //E/W longitude
z=Math.sin(ptA[0]);
x=Math.cos(ptA[0])*Math.cos(ptA[1]);
y=Math.cos(ptA[0])*Math.sin(ptA[1]);

return [x,y,z]; //a006200848
} //function LatitudeLongitude2xyz()





//arg1= vector A accurate direction
//arg2= vector B approximate direction
//arg3=1,2,3,4,5,6
//arg3='xz','xy','yx','yz','zy','zx'
//'xz' mean x axis=arg1=vector A
//'xz' mean z axis=arg2=vector B
//other are similar defined.
//
//function eyeUpMat() document
//please refer to
//http://freeman2.com/sphtrian.htm
//from [a name="docA_001"]
// to  [a name="docA_030"]
//a007082127
//
function eyeUpMat(arg1,arg2,arg3)
{ //a006181826

//vector A dot vector B
function AdB(A,B){return (A[0]*B[0]+A[1]*B[1]+A[2]*B[2]);}

//vector length
function L1(A){return Math.sqrt(A[0]*A[0]+A[1]*A[1]+A[2]*A[2]);}

//change string to array of string
function slic0(q){if(typeof(q)!='string')return q;if(q.indexOf(',')>0)q=q.split(',');else{while(q.indexOf('  ')>=0)q.replace('  ',' ');var i=0;var j=q.length-1;if(q.charAt(i)==' ')i++;if(q.charAt(j)==' ')j--;q=q.substring(i,j+1);q=q.split(' ');}return q;}//a006181845

var i,j,k,L;
var va=arg1;
if(typeof(va)=='string')va=slic0(va);
if(typeof(va)!='object'||va.length!=3)
return 'eyeUpMat() error 01'

var vb=arg2;
if(typeof(vb)=='string')vb=slic0(vb);
if(typeof(vb)!='object'||vb.length!=3)
return 'eyeUpMat() error 02'

L=L1(va);va=[va[0]/L,va[1]/L,va[2]/L]
L=L1(vb);vb=[vb[0]/L,vb[1]/L,vb[2]/L]

var f1=va[1]*vb[2]-va[2]*vb[1]
var f2=va[2]*vb[0]-va[0]*vb[2]
var f3=va[0]*vb[1]-va[1]*vb[0]
var absf=Math.sqrt(f1*f1+f2*f2+f3*f3);

if(Math.abs(absf)<1.e-8)
return 'eyeUpMat() error 03'

var vg=[];
vg[0]=f1/absf;
vg[1]=f2/absf;
vg[2]=f3/absf;

var vh=[];
vh[0]=va[1]*vg[2]-va[2]*vg[1]
vh[1]=va[2]*vg[0]-va[0]*vg[2]
vh[2]=va[0]*vg[1]-va[1]*vg[0]

var vm=[];
vm[0]=-vh[0]
vm[1]=-vh[1]
vm[2]=-vh[2]

//a006191911 begin correction

if(typeof(arg3)=='string') //a006192114 if()
arg3=arg3.toLowerCase();

if(arg3==1||arg3=='xz')
return [[va[0],vg[0],vh[0]],
[va[1],vg[1],vh[1]],
[va[2],vg[2],vh[2]]];
if(arg3==2||arg3=='xy')
return [[va[0],vm[0],vg[0]],
[va[1],vm[1],vg[1]],
[va[2],vm[2],vg[2]]];
if(arg3==3||arg3=='yx')
return [[vh[0],va[0],vg[0]],
[vh[1],va[1],vg[1]],
[vh[2],va[2],vg[2]]];
if(arg3==4||arg3=='yz')
return [[vg[0],va[0],vm[0]],
[vg[1],va[1],vm[1]],
[vg[2],va[2],vm[2]]];
if(arg3==5||arg3=='zy')
return [[vg[0],vh[0],va[0]],
[vg[1],vh[1],va[1]],
[vg[2],vh[2],va[2]]];
if(arg3==6||arg3=='zx')
return [[vm[0],vg[0],va[0]],
[vm[1],vg[1],va[1]],
[vm[2],vg[2],va[2]]];
//a006192022 done correction

return 'function eyeUpMat() can not handle input data.'

} //function eyeUpMat(arg1,arg2,arg3)




function drawSphereTriangle(
   arg1,arg2,arg3,arg4
  ,arg5) // a006171848 use arg5
{ //a006141523

//one argument for open blank drawing board.
//three arguments for given A,B,C three vectors.
// four arguments for given A,B two vectors and angles
// five arguments is four arguments plus reverse 
//a006171851
//none of above, stop
//
if(arguments.length!=3 
 &&arguments.length!=4 
 &&arguments.length!=5 
 &&arguments.length!=1 
  )return 'drawSphereTriangle() error 01'

//variable givenABC record input
//is three variables or four.
var givenABC=3; //a006170345
if(arguments.length!=3)givenABC=4

if(typeof(xyatL)=='function'
 &&typeof(QAxatL)=='object'
 &&typeof(QAyatL)=='object'
) //a006141632
xyatL(QAxatL,QAyatL);
if(typeof(xyaxis01)=='function'
 &&typeof(QAchkbxID51)=='object'
) 
xyaxis01(QAchkbxID51); 

QAspanAlert10.innerHTML='build data after 1st draw'
QAboxc11.value=
QAboxc12.value=
QAboxc13.value=
QAboxc14.value=
QAboxc15.value=
QAboxc16.value='';

var MyGraph = new XYGraph();
var MyLine1 = new XYLine();
var MyLine2 = new XYLine();
MyLine1.VMLpointshapetype="none";
MyLine2.VMLpointshapetype="none";

	MyGraph.ymax=QAboxYmax0.value;
 MyGraph.ymin=QAboxYmin0.value;
	MyGraph.xmax=QAboxXmax0.value;
 MyGraph.xmin=QAboxXmin0.value;
	MyGraph.gheight=QAboxHeight0.value;
 MyGraph.gwidth=QAboxWidth0.value;


MyLine1.x=[0,0];
MyLine1.y=[0,0];
MyGraph.Plot(MyLine1); 
if(arg1==0) 
  { //open blank drawing board.
QAdivGraph10.innerHTML = MyGraph; 
return;  //a006141533
  }

//all vector normalize.
//len0 record vector length.
var len0; 

var lineCWS=[]; //color, width, solidot
lineCWS[0]=[];
lineCWS[1]=[];
lineCWS[2]=[];
lineCWS[3]=[]; //a005171645
lineCWS[4]=[];
lineCWS[5]=[]; //a005182235

lineCWS[6]=[]; //a007051320
lineCWS[7]=[]; //a007061004

//following code depend on
//web page elements ID QA...
lineCWS[0][0]=QAxzclr.value
lineCWS[0][1]=QAxzwth.value
lineCWS[0][2]=QAxzsdd.value
lineCWS[1][0]=QAaclr.value
lineCWS[1][1]=QAawth.value
lineCWS[1][2]=QAasdd.value
lineCWS[2][0]=QAbclr.value
lineCWS[2][1]=QAbwth.value
lineCWS[2][2]=QAbsdd.value
lineCWS[3][0]=QAcclr.value
lineCWS[3][1]=QAcwth.value
lineCWS[3][2]=QAcsdd.value
//a005171746 lineCWS[4]
lineCWS[4][0]=QAabcclr.value
lineCWS[4][1]=QAabcwth.value
lineCWS[4][2]=QAabcsdd.value
//a005182232 lineCWS[5]
lineCWS[5][0]=QAaxsclr.value
lineCWS[5][1]=QAaxswth.value
lineCWS[5][2]=QAaxssdd.value

//a007051228 lineCWS[6]
lineCWS[6][0]=QAstarclr.value
lineCWS[6][1]=QAstarwth.value
lineCWS[6][2]=QAstarsdd.value

//a007061005 lineCWS[7]
lineCWS[7][0]=QAtanclr.value
lineCWS[7][1]=QAtanwth.value
lineCWS[7][2]=QAtansdd.value

//If input Chinese 紅色=red， 藍色=blue
//next if() change to '#FF0000'， '#0000FF'
if(typeof(cn2color)=='function')
  { //a006171532
for(i=0;i<lineCWS.length;i++)
lineCWS[i][0]=cn2color(lineCWS[i][0])
  }

//vector a and vector b 
//are from user input.
var a=cleanData(arg1)
var b=cleanData(arg2)

//Given A,B and angle, vector c is derived
//Given A,B,C three vectors. c is given
var c;

//Given A,B and angle, 
//angle variable
var rotAngA,rotAngB;

//vector has three numbers, 
//if not three, return error.
a=a.split(' ');
b=b.split(' ');
if(a.length!=3
 ||b.length!=3
  )return 'drawSphereTriangle() error 02'

//vector has three numbers, 
//if not number, return error.
if(isNaN(a[0])
 ||isNaN(a[1])
 ||isNaN(a[2])
  )return 'drawSphereTriangle() error 03'

if(isNaN(b[0])
 ||isNaN(b[1])
 ||isNaN(b[2])
  )return 'drawSphereTriangle() error 04'

//vector length can not be zero
//if zero, return error.
var L;
L=length1(a)
if(Math.abs(L)<1.e-6)
return 'drawSphereTriangle() error 05'
a=[a[0]/L,a[1]/L,a[2]/L]

L=length1(b)
if(Math.abs(L)<1.e-6)
return 'drawSphereTriangle() error 06'
b=[b[0]/L,b[1]/L,b[2]/L]

var alpha,beta,gamma;
var arcA,arcB,arcC;


if(givenABC==3) //a006170347
  { 
//Given A,B,C three vectors.
//read vector c here
c=cleanData(arg3)
c=c.split(' ');
  }
else //if(givenABC==4)
  { 
//Given A,B and angle, 
//derive vector c here

//a006160020
//vertex A angle in radian
rotAngA=arg3;
rotAngA=parseFloat(rotAngA)
rotAngA=rotAngA*Math.PI/180;

//vertex B angle in radian
rotAngB=arg4;
rotAngB=parseFloat(rotAngB)
rotAngB=rotAngB*Math.PI/180;
//what is next? //a006160029


/** a006160050
α=rotAngA/2
β=rotAngB/2
c=arc(A,B)
α,β,c all given, from
cos(γ)=-cos(α)*cos(β)
       +sin(α)*sin(β)*cos(c) [52c]
find γ 
after get γ from
spherical triangle sine law
sin(a)   sin(b)   sin(c)
------ = ------ = ------ [50]式
sin(α)   sin(β)   sin(γ)

that is
 sin(a)/sin(α)
=sin(b)/sin(β)
=sin(c)/sin(γ)

that is
 sin(arcA)/sin(alpha)
=sin(arcB)/sin(beta)
=sin(arcC)/sin(gamma)

find a =arc(B,C)
find b =arc(C,A)
get a, b then vector C 
can be calculated
a006160055

a006160900 think few hours,
given a,b ，how to find c?
just realize
Simon L. Altmann
Rotations, Quaternions, 
and Double Groups 
ISBN 0-486-44518-6
page 159 equation (15) is
answer (method 1)
a006160903 

rotation matrix is method 2
quaternion is method 3
on a006161811 choose method 3
/**/

//vector A rotation angle/2
//it is triangle vertex A angle
//alpha=rotAngA/2
//a007091339 drop "/2"
alpha=rotAngA;

//vector B rotation angle/2
//it is triangle vertex B angle
//beta=rotAngB/2
//a007091340 drop "/2"
beta=rotAngB;

with(Math) 
    { //a006160809

//a006161548 alpha beta gamma
//method 3 is
//from given A vector and angle/2
//find quaternion A
//from given B vector and angle/2
//find quaternion B
//use qC=quatquat(qA,qB) 
//find qC, then from qC 
//find vector C and angle C
//
var sinTmp

//create qA
sinTmp=sin(alpha)
var qA=[cos(alpha),a[0]*sinTmp,
       a[1]*sinTmp,a[2]*sinTmp]

//create qB
sinTmp=sin(beta)
var qB=[cos(beta),b[0]*sinTmp,
      b[1]*sinTmp,b[2]*sinTmp]

//find qC, use qC=quatquat(qA,qB) 
var qC;
if(arguments.length==4)
//qC=qA*qB
qC=quatquat(qA,qB); //a006161606 
else //arguments.length==5
//qC=qB*qA
qC=quatquat(qB,qA); //a006171855

//from qC find angle C vertex gamma
//gamma=acos(qC[0]);
//a007091354 correction
gamma=Math.PI-acos(qC[0]);

//from qC find vector C 
sinTmp=sin(gamma)
c=[qC[1]/sinTmp,
   qC[2]/sinTmp,
   qC[3]/sinTmp]
//given vector a and angle
//given vector b and angle
//find vector c. a006161622

    } //with(Math) 
  } //if(givenABC==3) else

if(c.length!=3
  )return 'drawSphereTriangle() error 07'
if(isNaN(c[0])
 ||isNaN(c[1])
 ||isNaN(c[2])
  )return 'drawSphereTriangle() error 08'
L=length1(c)
if(Math.abs(L)<1.e-6)
return 'drawSphereTriangle() error 09'
c=[c[0]/L,c[1]/L,c[2]/L]

//a006141545 stop
//a006141623 start

var i,j,k;

var MyLabel = new Label();
MyLabel.VMLpointshapetype="none";
MyLabel.labelcolor="black";

if(MyGraph.ymax-MyGraph.ymin<=3
 &&MyGraph.xmax-MyGraph.xmin<=3)
  { //a006230949 if()
MyLabel.x=-0.90;
MyLabel.y=-1.05;
MyLabel.label = "a006171242";
MyGraph.Drawlabel(MyLabel);
  }

///////////////


//loop has 360 degree
// 3 degree one step
var incAngle=3*Math.PI/180;
// 3 degree one step get 
//non-integer step0
//adjust incAngle to incrAng
//incAngle = 3 degree exactly
//incrAng = 3 degree - small error
//incrAng get integer step0
var incrAng;
var step0;

//diskArcAB record X loop
//every point x,y,z coord.
//
//diskArcAB[0][] is first point
// its x,y,z coord is diskArcAB[0][0] 
//diskArcAB[0][1] and diskArcAB[0][2] 

//radius
var diskRadi=1;
//a005191923

//center coord.
var diskCenter=[0,0,0];

//a006191513 start from bird body coord
//not start from ground coordinate
//in bird body coord x,y plane draw
//circle. perpendicular to bird z axis
//build rotation matrix rotMatAB, 
//rotMatBC, rotMatCA change loop 
//point coordinate 
//from bird body coord. measured value 
// to  ground coord. measured value 
//Loop point stay in physical space
//never move any.
//bird body coord. measured value is
//easy to calculate. Three matrix 
//convert to 
//loop one pass b,c two vectors
//loop two pass c,a two vectors
//loop three pass a,b two vectors 
//Three transform draw triangle 
//three sides, all be great circle.
var rotMatAB;
var rotMatBC;
var rotMatCA;
//matrix rotateMat rotate outer frame
//great circle, from bird body x,y 
//plane to a direction use eye line as
//circle axis. [0,0,0] to [Ix,Iy,Iz]
var rotateMat;

//a006191520 start
//puzzled a while, rotation matrix
//rotate start from where? and 
//rotate to where? Now make it clear.
//Why "draw circle on ground X,Y plane
//first" is error?
//a006191513 change "ground" 
//to "bird body"
//Why "draw circle on bird x,y plane
//first" is correct?
//*** Because after rotation, 
//*** all points are ground
//*** coordinate system value.
//therefore before rotation, easy
//to calculate circle must be on
//bird body coordinate x,y plane.
//Can not be on ground X,Y plane
//"rotate" from ground to ground 
//is non-sense.
//Transform must take place between
//ground coord. and bird body coord.
//a006191527 stop

//Draw great circle whole circle
var diskABarc
var diskBCarc
var diskCAarc

//Draw great circle partial circle
//triangle side only
var sideAB0; //a006151240
var sideBC0; 
var sideCA0; 

//diskABStr0 store 3D data
//then transform to 2D data
var diskABStr0
var diskBCStr0
var diskCAStr0

//ansdiskAB save 2D data
var ansdiskAB
var ansDiskBC
var ansDiskCA

//eye x,y,z coordinate and
// up xmymz components
//all 3D to 2D transform use
//same eye,up value
var Iview=s2aEZ(QAeyeview.value);
var Uview=s2aEZ(QA_upview.value);

var title0=''; //a005171903

//G = ground
var rotStrG=''  //ground x,y,z  //a005171701
+'1, 0, 0\n'
+'0, 0, 0\n'
+'0, 1, 0\n'
+'0, 0, 0\n'
+'0, 0, 1\n'

var ansG=run3Dto2D(rotStrG
,Iview,Uview);

//function XYGraphCode(inxy)
//has default line color, width,
//solidot. Next code change 
//default to user assigned value.
//
ansG[0]=ansG[0].replace('weight=1',
'weight='+lineCWS[0][1])
ansG[0]=ansG[0].replace('color=#000000',
'color='+lineCWS[0][0])
ansG[0]=ansG[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[0][2])

eval(ansG[0]); //a005171706


title0=''
+'//copy box11 code, paste to http://freeman2.com/xygraphe.htm\n'
+'////box1, click [%%Draw] xygraphe.htm let you build your page.\n'
+'//ground X,Y,Z triad.\n'
QAboxc11.value+=title0+ansG[0]+'\n\n';

title0=''
+'//ground X,Y,Z triad.\n'
QAboxc12.value+=title0+ansG[1]+'\n\n';
QAboxc13.value+=title0+rotStrG+'\n\n';

///////////////
//
//Above is ground X,Y,Z triad.
//Below is a,b,c three vectors

var rotStrABC='' //a,b,c three vectors
+a+'\n'
+'0,0,0\n'
+b+'\n'
+'0,0,0\n'
+c+'\n'
+'0,0,0\n'
//a006142009

//Base on eye x,y,z Iview and 
//up direction Uview
//change 3D data to 2D data
var ansABC=run3Dto2D(rotStrABC
,Iview,Uview);

//ansABC[0] has XYGraph code
//change default line color,
//width, solidot to user assigned
//value
//here is a,b,c three vectors
ansABC[0]=ansABC[0].replace('weight=1',
'weight='+lineCWS[4][1])
ansABC[0]=ansABC[0].replace('color=#000000',
'color='+lineCWS[4][0])
ansABC[0]=ansABC[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[4][2])

eval(ansABC[0]); //a005171748

title0='//a,b,c three vectors\n'
QAboxc11.value+=title0+ansABC[0]+'\n\n';
QAboxc12.value+=title0+ansABC[1]+'\n\n';
QAboxc13.value+=title0+rotStrABC+'\n\n';


////////////////////
//
//Above is a,b,c three vectors
//Below is a,b great circle

var diskArcAB=[];
var abAngl=Math.acos(a[0]*b[0]
 +a[1]*b[1]+a[2]*b[2]); //a006141743
//a,b great circle rotation matrix
rotMatAB=eyeSideUp(a,b); //a006151158
//Circle pass vector a and pass 
//vector b , this circle is called
//a,b circle.
//a,b circle made up with 120 broken
//lines, some of them are triangle
//side. Variable stepab record 
//number of SIDE broken lines. When
//build XYGraph code, draw whole 
//circle (light color thin line)
//and draw triangle side (red thick)
//Three loops and three sides all
//use same method.
//a007091036 translate to here
//
stepab=abAngl/incAngle;
stepab=parseInt(stepab)+1;
incrAng=abAngl/stepab
for(i=0;i<=stepab;i++)
    { 
//draw circle on bird body x,y plane first
diskArcAB[i]=[];
diskArcAB[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcAB[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcAB[i][2]=0; 
//circle z coordinate is zero, easy work.
    }

//above draw a,b circle a,b side 
//i=0;i<=stepab part of triangle side
//following draw a,b whole circle
//i=stepab;i<=k
//triangle side and great circle
//have different color/width/solidot
k=Math.PI*2/incrAng
k=parseInt(k)+1;
for(i=stepab;i<=k;i++)
    {
diskArcAB[i]=[];
diskArcAB[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcAB[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcAB[i][2]=0;
    }

//
//transform a,b circle from circle
//perpendicular to bird body z axis
//to circle pass vector a and b.
//In this application, parallel move
//amount is [0,0,0]
diskABarc=rotateShift(
//transpoMat(rotMatAB), //rot.matrix a006152208
//a006191254 from rotMatAB del transpoMat()
//a006191254 think why delete transpoMat() 
//get correct answer? see doc at a006191520
rotMatAB, //rot.matrix a006152208
//transpoMat(diskArcAB), //all points
//a006191249 from diskArcAB del transpoMat()
diskArcAB, //all points on circle
diskCenter,   // [0,0,0]
[0,0,0]) //parallel move amount

//change string matrix to 
//number array matrix
diskABarc=s2aEZ(diskABarc);

//diskABcircle=transpoMat(diskABarc)
//a006191320 del transpoMat()
diskABcircle=diskABarc

//function dim3to2b() require input
//inStr0 be one line one point x,y,z 
//do not allow one line many points.
//a005191548  
//
diskABStr0=''//a005191549
for(i=0;i<diskABcircle.length;i++)
diskABStr0+=diskABcircle[i]+'\n';

ansdiskAB=run3Dto2D(diskABStr0
      ,Iview,Uview); //a005191550

//a,b circle line color/wodth/solidot
ansdiskAB[0]=ansdiskAB[0].replace('weight=1',
'weight='+lineCWS[3][1].charAt(0))
ansdiskAB[0]=ansdiskAB[0].replace('color=#000000',
'color='+lineCWS[3][0])
ansdiskAB[0]=ansdiskAB[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[3][2])

//All neat draw page output XYGraph 
//code. all use eval( XYGraph code )
//make sure XYGraph code is OK.
eval(ansdiskAB[0]); //a005182320

//ansdiskAB[0] has XYGraph code
//eval(ansdiskAB[0]) IS draw command
//ansdiskAB[0] also output to Box11
//Above structure make sure XYGraph 
//code in Box11 is a working version
//
title0='//a,b loop all points\n'
QAboxc11.value+=title0+ansdiskAB[0]+'\n\n';
QAboxc12.value+=title0+ansdiskAB[1]+'\n\n';
QAboxc13.value+=title0+diskABStr0+'\n';
//Above is a,b loop all points
//below is a,b circle triangle side
//from a,b circle print first stepab+1 
//points.

sideAB0=''//a006151242
for(i=0;i<=stepab;i++)
sideAB0+=diskABcircle[i]+'\n';
ansSideAB=run3Dto2D(sideAB0
      ,Iview,Uview); //a006151244
//a,b side line color,width,solidot
ansSideAB[0]=ansSideAB[0].replace('weight=1',
'weight='+lineCWS[5][1].charAt(0))
ansSideAB[0]=ansSideAB[0].replace('color=#000000',
'color='+lineCWS[5][0])
ansSideAB[0]=ansSideAB[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[5][2])

eval(ansSideAB[0]); //a006151246
title0='//a,b circle triangle side\n'
QAboxc11.value+=title0+ansSideAB[0]+'\n\n';
QAboxc12.value+=title0+ansSideAB[1]+'\n\n';
QAboxc13.value+=title0+diskABStr0+'\n';

//a007091059 translate stop
//a007091220 translate start

////////////////////
//
//above is a,b circle
//below is b,c circle

var diskArcBC=[];
var bcAngl=Math.acos(b[0]*c[0]
 +b[1]*c[1]+b[2]*c[2]); //a006151538
//b,c circle rotation matrix
rotMatBC=eyeSideUp(b,c); //a006151158
stepbc=bcAngl/incAngle;
stepbc=parseInt(stepbc)+1;
incrAng=bcAngl/stepbc
for(i=0;i<=stepbc;i++)
    { 
diskArcBC[i]=[];
diskArcBC[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcBC[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcBC[i][2]=0;
    }
k=Math.PI*2/incrAng
k=parseInt(k)+1;
for(i=stepbc;i<=k;i++)
    {
diskArcBC[i]=[];
diskArcBC[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcBC[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcBC[i][2]=0;
    }

diskBCarc=rotateShift(
rotMatBC, //rotation matrix a006152209
diskArcBC,
diskCenter,
[0,0,0])
diskBCarc=s2aEZ(diskBCarc);

diskBCcircle=diskBCarc

diskBCStr0=''//a005191549
for(i=0;i<diskBCcircle.length;i++)
diskBCStr0+=diskBCcircle[i]+'\n';
ansdiskBC=run3Dto2D(diskBCStr0
      ,Iview,Uview); //a005191550
//b,c loop line color,width,solidot
ansdiskBC[0]=ansdiskBC[0].replace('weight=1',
'weight='+lineCWS[1][1].charAt(0))
ansdiskBC[0]=ansdiskBC[0].replace('color=#000000',
'color='+lineCWS[1][0])
ansdiskBC[0]=ansdiskBC[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[1][2])
eval(ansdiskBC[0]); //a005182320
title0='//b,c loop all points\n'
QAboxc11.value+=title0+ansdiskBC[0]+'\n\n';
QAboxc12.value+=title0+ansdiskBC[1]+'\n\n';
QAboxc13.value+=title0+diskBCStr0+'\n';
//Above is b,c loop all points
//below is b,c circle triangle side
//from b,c circle print first stepbc+1 
//points.
sideBC0=''//a006151242
for(i=0;i<=stepbc;i++)
sideBC0+=diskBCcircle[i]+'\n';
ansSideBC=run3Dto2D(sideBC0
      ,Iview,Uview); //a006151244
ansSideBC[0]=ansSideBC[0].replace('weight=1',
'weight='+lineCWS[5][1].charAt(0))
ansSideBC[0]=ansSideBC[0].replace('color=#000000',
'color='+lineCWS[5][0])
ansSideBC[0]=ansSideBC[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[5][2])
eval(ansSideBC[0]); //a006151246
title0='//b,c circle triangle side\n'
QAboxc11.value+=title0+ansSideBC[0]+'\n\n';
QAboxc12.value+=title0+ansSideBC[1]+'\n\n';
QAboxc13.value+=title0+diskBCStr0+'\n';


////////////////////
//
//above is b,c circle
//below is c,a circle

var diskArcCA=[];
var caAngl=Math.acos(c[0]*a[0]
 +c[1]*a[1]+c[2]*a[2]); //a006151546
//c,a circle rotation matrix
rotMatCA=eyeSideUp(c,a); 
stepca=caAngl/incAngle;
stepca=parseInt(stepca)+1;
incrAng=caAngl/stepca
for(i=0;i<=stepca;i++)
    { 
diskArcCA[i]=[];
diskArcCA[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcCA[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcCA[i][2]=0;
    }
k=Math.PI*2/incrAng
k=parseInt(k)+1;
for(i=stepca;i<=k;i++)
    {
diskArcCA[i]=[];
diskArcCA[i][0]=diskRadi*Math.cos(i*incrAng)
diskArcCA[i][1]=diskRadi*Math.sin(i*incrAng)
diskArcCA[i][2]=0;
    }

diskCAarc=rotateShift(
rotMatCA, //rotation matrix a006152210
diskArcCA,
diskCenter,
[0,0,0])
diskCAarc=s2aEZ(diskCAarc);

diskCAcircle=diskCAarc

diskCAStr0=''//a005191549
for(i=0;i<diskCAcircle.length;i++)
diskCAStr0+=diskCAcircle[i]+'\n';
ansdiskCA=run3Dto2D(diskCAStr0
      ,Iview,Uview); //a005191550
//c,a circle line color/wodth/solidot
ansdiskCA[0]=ansdiskCA[0].replace('weight=1',
'weight='+lineCWS[2][1].charAt(0))
ansdiskCA[0]=ansdiskCA[0].replace('color=#000000',
'color='+lineCWS[2][0])
ansdiskCA[0]=ansdiskCA[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[2][2])
eval(ansdiskCA[0]); //a005182320
title0='//c,a loop all points\n'
QAboxc11.value+=title0+ansdiskCA[0]+'\n\n';
QAboxc12.value+=title0+ansdiskCA[1]+'\n\n';
QAboxc13.value+=title0+diskCAStr0+'\n';
//Above is c,a loop all points
//below is c,a circle triangle side
//from c,a circle print first stepca+1 
//points.

sideCA0=''//a006151242
for(i=0;i<=stepca;i++)
sideCA0+=diskCAcircle[i]+'\n';
ansSideCA=run3Dto2D(sideCA0
      ,Iview,Uview); //a006151244
ansSideCA[0]=ansSideCA[0].replace('weight=1',
'weight='+lineCWS[5][1].charAt(0))
ansSideCA[0]=ansSideCA[0].replace('color=#000000',
'color='+lineCWS[5][0])
ansSideCA[0]=ansSideCA[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[5][2])
eval(ansSideCA[0]); //a006151246
title0='//c,a circle triangle side\n'
QAboxc11.value+=title0+ansSideCA[0]+'\n\n';
QAboxc12.value+=title0+ansSideCA[1]+'\n\n';
QAboxc13.value+=title0+diskCAStr0+'\n';




if(givenABC==4&&QAchkbxID72.checked)
  {
////////////////////
//
//above is c,a circle
//below is l,m,n aux. vector a007051233

var l=a;
var m=b;
var p=v1Xv2(a,b);
var lStar,mStar,pStar,n;

//tute0055.htm eqn.rod136
var mp=v1Xv2(m,p);
var lmp=l[0]*mp[0]+l[1]*mp[1]+l[2]*mp[2];
lStar=[mp[0]/lmp,mp[1]/lmp,mp[2]/lmp];


//tute0055.htm eqn.rod138 but drop "‧n"
var pl=v1Xv2(p,l);
mStar=[pl[0]/lmp,pl[1]/lmp,pl[2]/lmp];
//if not drop "‧n"， two vectors dot 
//result is a scalar, non-vector.
//here need vector part. But this
//vector was not listed in tutc0055.htm

//tute0055.htm eqn.rod139 but drop "‧n"
var lm=v1Xv2(l,m);
pStar=[lm[0]/lmp,lm[1]/lmp,lm[2]/lmp];

var rotStrLmn='' //l*,m*,p* three vectors
+lStar+'\n'
+'0,0,0\n'
+mStar+'\n'
+'0,0,0\n'
+pStar+'\n'
+'0,0,0\n'
// +n+'\n'
//a007051311

//Base on eye x,y,z Iview and 
//up direction Uview
//change 3D data to 2D data
var ansLmn=run3Dto2D(rotStrLmn
,Iview,Uview);

//change default line color,
//width, solidot to user assigned
//value
//here is l*,m*,p* three vectors
ansLmn[0]=ansLmn[0].replace('weight=1',
'weight='+lineCWS[6][1])
ansLmn[0]=ansLmn[0].replace('color=#000000',
'color='+lineCWS[6][0])
ansLmn[0]=ansLmn[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[6][2])

eval(ansLmn[0]);

title0='//l*,m*,p* three vectors\n'
QAboxc11.value+=title0+ansLmn[0]+'\n\n';
QAboxc12.value+=title0+ansLmn[1]+'\n\n';
QAboxc13.value+=title0+rotStrLmn+'\n\n';
//a007051314
  } //if(givenABC==4&&QAchkbxID72.checked)





////////////////////
//
//above is l*,m*,p* three vectors
//below is outer frame circle

//outer frame circle rot.matrix 
//a006151900 
var accurateZ=eyeXup(Iview,Uview);
rotateMat=eyeSideUp(accurateZ,
    vec1(v1Xv2(accurateZ,Iview)))

var diskRimArray=[]; //a006151913
rimAngl=Math.PI*2; //a006151831
step0=rimAngl/incAngle;
step0=parseInt(step0)+1;
incrAng=rimAngl/step0
for(i=0;i<=step0;i++)
    { 
diskRimArray[i]=[];
diskRimArray[i][0]=diskRadi*Math.cos(i*incrAng)
diskRimArray[i][1]=diskRadi*Math.sin(i*incrAng)
diskRimArray[i][2]=0;
    }

diskRimarc=rotateShift(
rotateMat, //rotation matrix a006152211
diskRimArray,
diskCenter,
[0,0,0])
diskRimarc=s2aEZ(diskRimarc);

diskRimcircle=diskRimarc

diskRimStr0=''//a005191549
for(i=0;i<diskRimcircle.length;i++)
diskRimStr0+=diskRimcircle[i]+'\n';
ansdiskRim=run3Dto2D(diskRimStr0
      ,Iview,Uview); //a005191550
//outer frame circle line color,
//width,solidot.
ansdiskRim[0]=ansdiskRim[0].replace('weight=1',
'weight='+lineCWS[0][1].charAt(0))
ansdiskRim[0]=ansdiskRim[0].replace('color=#000000',
'color='+lineCWS[0][0])
ansdiskRim[0]=ansdiskRim[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[0][2])
eval(ansdiskRim[0]); //a005182320
title0='//outer frame circle points\n'
QAboxc11.value+=title0+ansdiskRim[0]+'\n';
QAboxc12.value+=title0+ansdiskRim[1]+'\n';
QAboxc13.value+=title0+diskRimStr0+'\n';


///////////////


//a006162105
var vertex1=a+'\n'+b+'\n'+c+'\n';
var vertex2=run3Dto2D(vertex1
      ,Iview,Uview);


var vertex3=s2aEZ(vertex2[1])
var labelStr0=''
+'var MyLabel = new Label();\n'
+'MyLabel.pointfillcolor="'+lineCWS[1][0]+'";\n'
+'MyLabel.x=0;\n'
+'MyLabel.y=0;\n'
+'MyLabel.VMLpointshapetype="'+'square'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'
+'MyLabel.x='+vertex3[0][0]+';\n'
+'MyLabel.y='+vertex3[0][1]+';\n'
+'MyLabel.VMLpointshapetype="'+'triangle'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'
+'MyLabel.x='+vertex3[1][0]+';\n'
+'MyLabel.y='+vertex3[1][1]+';\n'
+'MyLabel.VMLpointshapetype="'+'diamond'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'
+'MyLabel.x='+vertex3[2][0]+';\n'
+'MyLabel.y='+vertex3[2][1]+';\n'
+'MyLabel.VMLpointshapetype="'+'circle'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'


if(givenABC==4&&QAchkbxID72.checked)
  { //a007051526

var ansLmn1=s2aEZ(ansLmn[1]); //a007051552

labelStr0+=''
+'//l*,m*,p* three vectors l*=▲,m*=◆,p*=× a007051538\n'
+'MyLabel.x='+ansLmn1[0][0]+';\n'
+'MyLabel.y='+ansLmn1[0][1]+';\n'
+'MyLabel.VMLpointshapetype="'+'triangle'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'
+'MyLabel.x='+ansLmn1[2][0]+';\n'
+'MyLabel.y='+ansLmn1[2][1]+';\n'
+'MyLabel.VMLpointshapetype="'+'diamond'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n'
+'MyLabel.x='+ansLmn1[4][0]+';\n'
+'MyLabel.y='+ansLmn1[4][1]+';\n'
//+'MyLabel.VMLpointshapetype="'+'circle'+'";\n'
//a007051648 change to 'x'
+'MyLabel.VMLpointshapetype="'+'x'+'";\n'
+'MyGraph.Drawlabel(MyLabel);\n' 

  } //if(givenABC==4&&QAchkbxID72.checked)


eval(labelStr0) //a006162123

title0='Vertex A:▲ ; Vertex B:◆ ; Vertex C:● ; [0,0,0]:■\n'
QAboxc11.value+='//'+title0+labelStr0+'\n';
QAspanEqnDoc10.innerHTML=title0;//a006162125


//Liu,Hsinhan sign here
QAboxc11.value+=''
+'/***** Data created on '+yearMdhm()+' *****/\n' //a007071614
+'/***** Spherical triangle #37995#28450;a006152047 *****/'
+'\n\n';


if(givenABC==3) //given A,B,C three pt.
  {
var a9=bcAngl; //a906152250
var b9=caAngl;
var c9=abAngl;

//three vertex angles in rad.
with(Math) 
    {
alpha=acos((cos(a9)-cos(b9)*cos(c9))/(sin(b9)*sin(c9)))
beta =acos((cos(b9)-cos(c9)*cos(a9))/(sin(c9)*sin(a9)))
gamma=acos((cos(c9)-cos(a9)*cos(b9))/(sin(a9)*sin(b9)))
    } //a906152257
//a906152305 
//Spherical triangle cosine law 
//calculation result look OK.
  } //given A,B,C three pt.

QAboxc14.value+=''
+'vec.A inp='+arg1+'\n'
+'vec.A one='+a+'\n'
+'vec.B inp='+arg2+'\n'
+'vec.B one='+b+'\n'
+(givenABC==3?'vec.C inp='+arg3+'\n':'')
+'vec.C one='+c+'\n'
+'verify C.one='+(c[0]*c[0]+c[1]*c[1]+c[2]*c[2])+'\n'
+'AB angle abAngl，rad:'+abAngl+'\n'
+'AB angle abAngl，deg:'+(abAngl*180/Math.PI)+'\n'
+'BC angle bcAngl，rad:'+bcAngl+'\n'
+'BC angle bcAngl，deg:'+(bcAngl*180/Math.PI)+'\n'
+'CA angle caAngl，rad:'+caAngl+'\n'
+'CA angle caAngl，deg:'+(caAngl*180/Math.PI)+'\n'
+'vertex A angle rad:'+alpha+'\n'
+'vertex A angle deg:'+(alpha*180/Math.PI)+'\n'
+'vertex B angle rad:'+beta+'\n'
+'vertex B angle deg:'+(beta*180/Math.PI)+'\n'
+'vertex C angle rad:'+gamma+'\n' //a006152258
+'vertex C angle deg:'+(gamma*180/Math.PI)+'\n'
+'∠A+∠B+∠C-180='+((alpha+beta+gamma-Math.PI)*180/Math.PI)
+' degree\n' //a007091347 add ∠A+∠B+∠C-180
//a006221918 add area and normal
+'spherical triangle area='+(alpha+beta+gamma-Math.PI)*1*1+'\n'
+'AB normal='+vec1(v1Xv2(a,b))+'\n'
+'BC normal='+vec1(v1Xv2(b,c))+'\n'
+'CA normal='+vec1(v1Xv2(c,a))+'\n'
+'eye x,y,z input:'+QAeyeview.value+'\n'
+' up x,y,z input:'+QA_upview.value+'\n'
+'a006160932\n'


var tangentInfo=''; //a006221444

if(typeof(QAchkbxID61)=='object'
 &&QAchkbxID61.checked)
  { //a006221203
var pDE=draw2tangents(a,b,c)

var ptD=pDE[0];
var ptE=pDE[1];

//Vertex A two tangents a006221208
var A2tan='' 
+b+'\n'
+ptD+'\n'
+a+'\n'
+ptE+'\n'
+c+'\n'
+ptE+'\n'
+ptD+'\n'
//a006221214

//Base on eye x,y,z Iview and 
//up direction Uview
//change 3D data to 2D data
var ansA2tan=run3Dto2D(A2tan
,Iview,Uview);

tangentInfo+=''
+'Vertex A two tangents, 3D data //a006221446\n'
+A2tan+'\n'
+'eye x,y,z Iview=\n'
+Iview+'\n'
+'Vertex A two tangents, 2D data \n'
+ansA2tan+'\n'

//ansA2tan[0] has XYGraph code
//change default line color,
//width, solidot to user assigned
//value
//here is Vertex A two tangents 
ansA2tan[0]=ansA2tan[0].replace('weight=1',
'weight='+lineCWS[7][1])
ansA2tan[0]=ansA2tan[0].replace('color=#000000',
'color='+lineCWS[7][0])
ansA2tan[0]=ansA2tan[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[7][2])
//a007061007 Vertex A two tangents 
//from lineCWS[4] to lineCWS[7]

eval(ansA2tan[0]); 

title0='//Vertex A two tangents\n'
QAboxc11.value+=title0+ansA2tan[0]+'\n\n';
QAboxc12.value+=title0+ansA2tan[1]+'\n\n';
QAboxc13.value+=title0+A2tan+'\n\n';

  } //if(...&&QAchkbxID61.checked)



if(typeof(QAchkbxID62)=='object'
 &&QAchkbxID62.checked)
  { //a006221426

var pDE=draw2tangents(b,c,a)
var ptD=pDE[0];
var ptE=pDE[1];

//Vertex B two tangents
var A2tan='' 
+c+'\n'
+ptD+'\n'
+b+'\n'
+ptE+'\n'
+a+'\n'
+ptE+'\n'
+ptD+'\n'

//Base on eye x,y,z Iview and 
//up direction Uview
//change 3D data to 2D data
var ansA2tan=run3Dto2D(A2tan
,Iview,Uview);

tangentInfo+=''
+'Vertex B two tangents, 3D data //a006221456\n'
+A2tan+'\n'
+'eye x,y,z Iview=\n'
+Iview+'\n'
+'Vertex B two tangents, 2D data\n'
+ansA2tan+'\n'

//ansA2tan[0] has XYGraph code
//change default line color,
//width, solidot to user assigned
//value
//here is Vertex B two tangents 
ansA2tan[0]=ansA2tan[0].replace('weight=1',
'weight='+lineCWS[7][1])
ansA2tan[0]=ansA2tan[0].replace('color=#000000',
'color='+lineCWS[7][0])
ansA2tan[0]=ansA2tan[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[7][2])

eval(ansA2tan[0]); 

title0='//Vertex B two tangents\n'
QAboxc11.value+=title0+ansA2tan[0]+'\n\n';
QAboxc12.value+=title0+ansA2tan[1]+'\n\n';
QAboxc13.value+=title0+A2tan+'\n\n';
//a006321432
  } //if(...&&QAchkbxID62.checked)




if(typeof(QAchkbxID63)=='object'
 &&QAchkbxID63.checked)
  { //a006321434

var pDE=draw2tangents(c,a,b)
var ptD=pDE[0];
var ptE=pDE[1];

//Vertex C two tangents 
var A2tan='' 
+a+'\n'
+ptD+'\n'
+c+'\n'
+ptE+'\n'
+b+'\n'
+ptE+'\n'
+ptD+'\n'

//Base on eye x,y,z Iview and 
//up direction Uview
//change 3D data to 2D data
var ansA2tan=run3Dto2D(A2tan
,Iview,Uview);

tangentInfo+=''
+'Vertex C two tangents 3D data //a006221458\n'
+A2tan+'\n'
+'eye x,y,z Iview=\n'
+Iview+'\n'
+'Vertex C two tangents 2D data \n'
+ansA2tan+'\n'

//ansA2tan[0] has XYGraph code
//change default line color,
//width, solidot to user assigned
//value
//here is Vertex C two tangents 
ansA2tan[0]=ansA2tan[0].replace('weight=1',
'weight='+lineCWS[7][1])
ansA2tan[0]=ansA2tan[0].replace('color=#000000',
'color='+lineCWS[7][0])
ansA2tan[0]=ansA2tan[0].replace('dashstyle=solid',
'dashstyle='+lineCWS[7][2])

eval(ansA2tan[0]); 

title0='//Vertex C two tangents\n'
QAboxc11.value+=title0+ansA2tan[0]+'\n\n';
QAboxc12.value+=title0+ansA2tan[1]+'\n\n';
QAboxc13.value+=title0+A2tan+'\n\n';

  } //if(...&&QAchkbxID63.checked)


QAboxc16.value=''
+tangentInfo //a006221446 
+QAboxc16.value


//calculate triangle center 
//x,y,z coordinate
QAspanAlert10.innerHTML='' //a006162148
+dotPrecision((a[0]+b[0]+c[0])/3)+','
+dotPrecision((a[1]+b[1]+c[1])/3)+','
+dotPrecision((a[2]+b[2]+c[2])/3)

QAboxc15.value+='' //a006170358
+'AB circle rotation matrix\n'
+mat2str(transpoMat(rotMatAB))
+'\n'
+'BC circle rotation matrix\n'
+mat2str(transpoMat(rotMatBC))
+'\n'
+'CA circle rotation matrix\n'
+mat2str(transpoMat(rotMatCA))
+'\n'
+'outer frame circle rotation matrix\n'
+mat2str(transpoMat(rotateMat))
+'\n'


QAdivGraph10.innerHTML = MyGraph; 


 //a006141523 start drawSphereTriangle(arg1) 
 //a006171242 done
}//function drawSphereTriangle(arg1) 


//開啟空白畫板
drawSphereTriangle(0); 






/**
2011-06-21-23-07 start download
http://lgdata.s3-website-us-east-1.amazonaws.com/docs/1336/243192/19770-pdf.pdf
2011-06-21-23-15 done download
06/21/2011  11:15 PM  808,287 

2011-06-22-10-12 open
spherical_trigonometry_lgdata.s3-website-us-east-1.amazonaws.com_19770-pdf.pdf
page 27/189 has spherical 
triangle graph.
Use 2D triangle cosine law
prove 3D triangle cosine law
Good method, simple and easy
to understand.
2011-06-22-10-19

2011-06-22-11-05 write code
draw spherical triangle and 
plane triangle relation. 
This is a graph in
spherical_trigonometry_lgdata.s3-website-us-east-1.amazonaws.com_19770-pdf.pdf
page 27/189 

/**/
function draw2tangents(arg1,arg2,arg3)
{ //a006221127
//change string data to 
//array of string data
function slic0(q){if(typeof(q)!='string')return q;if(q.indexOf(',')>0)q=q.split(',');else{while(q.indexOf('  ')>=0)q.replace('  ',' ');var i=0;var j=q.length-1;if(q.charAt(i)==' ')i++;if(q.charAt(j)==' ')j--;q=q.substring(i,j+1);q=q.split(' ');}return q;}//a006181845
//vector length
function L1(A){return Math.sqrt(A[0]*A[0]+A[1]*A[1]+A[2]*A[2]);}
//vector A dot vector B
function AdB(A,B){return (A[0]*B[0]+A[1]*B[1]+A[2]*B[2]);}
//a006181958 3x3 matrix A multiply 3x1 vector B
function mAvB(A,B){
return [A[0][0]*B[0]+A[0][1]*B[1]+A[0][2]*B[2],
A[1][0]*B[0]+A[1][1]*B[1]+A[1][2]*B[2],
A[2][0]*B[0]+A[2][1]*B[1]+A[2][2]*B[2]];
}
if(arguments.length!=3)
return 'draw2tangents() error 01';


var ptA=arg1;
var ptB=arg2;
var ptC=arg3;

if(typeof(ptA)=='string'
 &&typeof(ptB)=='string'
 &&typeof(ptC)=='string')
  {
ptA=slic0(ptA);
ptB=slic0(ptB);
ptC=slic0(ptC);
  } //if(typeof(ptA)=='string' && &&

if(ptA.length!=3||ptB.length!=3
 ||ptC.length!=3)
return 'draw2tangents() error 02';


var i,j,k;
for(i=0;i<3;i++)
  {
ptA[i]=parseFloat(ptA[i])
ptB[i]=parseFloat(ptB[i])
ptC[i]=parseFloat(ptC[i])
if(isNaN(ptA[i])||isNaN(ptB[i])
 ||isNaN(ptC[i]))
return 'draw2tangents() error 03';
  }


var L;
L=L1(ptA);
if(Math.abs(L)<1.e-8)
return 'draw2tangents() error 04';
ptA=[ptA[0]/L,ptA[1]/L,ptA[2]/L]
L=L1(ptB);
if(Math.abs(L)<1.e-8)
return 'draw2tangents() error 05';
ptB=[ptB[0]/L,ptB[1]/L,ptB[2]/L]
L=L1(ptC);
if(Math.abs(L)<1.e-8)
return 'draw2tangents() error 06';
ptC=[ptC[0]/L,ptC[1]/L,ptC[2]/L]


var q,r,s,t,u,v;
var a,b,c,d,e;
var ab,ac;
with(Math) //a006221228
  {
//a006221536
//next code erase QAboxc11.value
var rotMatAB=eyeUpMat(ptB,ptA,'xy')

var transpAB=transpoMat(rotMatAB); //a006221226

//a006221300
ab=
a=mAvB(transpAB,ptA);
b=mAvB(transpAB,ptB);
d=[]

q=AdB(a,b);
if(abs(q)<1.e-8)
return 'draw2tangents() error 07';

r=acos(q);
d[0]=1/cos(r);
//a006221320 d[1] correction
d[1]=0;
d[2]=0;
var ptD=mAvB(rotMatAB,d)

var rotMatAC=eyeUpMat(ptC,ptA,'xy')
var transpAC=transpoMat(rotMatAC); //a006221237
ac=
a=mAvB(transpAC,ptA);
c=mAvB(transpAC,ptC);
e=[]
q=AdB(a,c);
if(abs(q)<1.e-8)
return 'draw2tangents() error 08';
r=acos(q);
e[0]=1/cos(r);
//e[1]=sin(r);
//a006221321 e[1] correction
e[1]=0;
e[2]=0;
var ptE=mAvB(rotMatAC,e)
  } //with(Math)


QAboxc16.value='' 
+'A='+ptA+'\n'
+'B='+ptB+'\n'
+'C='+ptC+'\n'
+'D='+ptD+'\n'
+'E='+ptE+'\n'
+'===\n'
+'a='+ab+'\n'
+'b='+b+'\n'
+'d='+d+'\n'
+'oa dot ad='+(a[0]*(d[0]-a[0])+a[1]*(d[1]-a[1]))+'\n'
+'===\n'
+'a='+ac+'\n'
+'c='+c+'\n'
+'e='+e+'\n'
+'===\n'
+'rotMatAB=\n'+rotMatAB+'\n'
+'rotMatAC=\n'+rotMatAC+'\n'
+'a006221308dbg\n'


return [ptD,ptE]; //a006221240

} //function draw2tangents()
//a006221305





//2011-06-16-14-31 open
//math9710_unicode.htm
//2011-06-16-14-35 use pq(p,q)
//
function pq(p,q){//9710191047
var u=[0,0,0,0]; //9710191100
u[0]=p[0]*q[0]-(p[1]*q[1]+p[2]*q[2]+p[3]*q[3]);
u[1]=p[2]*q[3]-p[3]*q[2]+p[0]*q[1]+q[0]*p[1];
u[2]=p[3]*q[1]-p[1]*q[3]+p[0]*q[2]+q[0]*p[2];
u[3]=p[1]*q[2]-p[2]*q[1]+p[0]*q[3]+q[0]*p[3];
return u; //9710191102
}



//a006161448
//function quatquat(arg1,arg2) 
//input arg1,arg2 are two quaternion
//function quatquat(arg1,arg2) 
//do not calculate quat multiply quat
//quatquat() verify input are valid
//quaternion. If true, call pq(p,q)
//for quat multiply quat
//a006161450
//
function quatquat(arg1,arg2)
{ //a006161437
var q1=arg1;
var q2=arg2;
if(typeof(q1)=='string')
  {
q1=cleanData(q1);
q1=q1.split(' ');
  }
if(typeof(q2)=='string')
  {
q2=cleanData(q2);
q2=q2.split(' ');
  }

if(q1.length!=4)
return 'quatquat() error 01'
if(q2.length!=4)
return 'quatquat() error 02'
var i,j,k;
j=k=0;
for(i=0;i<4;i++)
  {
q1[i]=parseFloat(q1[i])
q2[i]=parseFloat(q2[i])
if(isNaN(q1[i]))
return 'quatquat() error 03'
if(isNaN(q2[i]))
return 'quatquat() error 04'
j+=q1[i]*q1[i];
k+=q2[i]*q2[i];
  }

if(Math.abs(j-1)>1.e-8)
return 'quatquat() error 05'
if(Math.abs(k-1)>1.e-8)
return 'quatquat() error 06'

return pq(q1,q2);
//a006161447

} //function quatquat(arg1,arg2)




//a006151122 function eyeSideUp
//eyeSideUp(arg1,arg2,arg3)
//
//function eyeXup(eye0,up0)
//eyeXup() set eye to be 
//after rotation accurate x axis
//eyeXup() set  up to be 
//after rotation approximate z axis
//
//function eyeSideUp() is different
//eyeSideUp() set eye to be 
//after rotation accurate x axis
//eyeSideUp() set  up to be 
//after rotation approximate y axis
//
//If draw 3D graph, need eyeXup()
//If draw spherical triangle, 
//need eyeSideUp()
//
//spherical triangle one side is
//partial of a great circle.
//To draw great circle, do it
//easier at bird body coordinate 
//system xy plane. its z coordinate
//is zero (easy part here) 
//Triangle side two end vectors,
//vec.a and vec.b
//Bird xy plane x axis coincide
//with vec.a
//Bird xy plane y axis stay in
//vec.a and vec.b plane.
//vec.a and vec.b are given.
//After rotation accurate x axis 
//direction is known
//After rotation approximate y 
//axis  direction is known
//Consider x axis be eye vector
//Consider y axis be  up vector
//Therefore 
//eyeSideUp(arg1,arg2,arg3)
//output rotation matrix for
//accurate x axis and
//approximate y axis (not z axis)
//a006151140
//
//a007091733 
//2011-06-15-11-41 write eyeSideUp()
//2011-06-18-18-26 write eyeUpMat()
//function eyeUpMat() is more 
//general than eyeSideUp()
//Liu,Hsinhan will use eyeUpMat()
//in the future page, not eyeSideUp()
//a007091738 
//
//
function eyeSideUp(arg1,arg2,arg3)
{ //a006151141
var eye0=arg1;
var sid0=arg2;

if(typeof(eye0)=='string')
  {
eye0=cleanData(eye0);
eye0=eye0.split(' ');
  }

if(typeof(sid0)=='string')
  {
sid0=cleanData(sid0);
sid0=sid0.split(' ');
  }

if(eye0.length!=3)
return 'eyeSideUp() error 01'
if(sid0.length!=3)
return 'eyeSideUp() error 02'
if(isNaN(eye0[0])||isNaN(eye0[1])
 ||isNaN(eye0[2]))
return 'eyeSideUp() error 03'
if(isNaN(sid0[0])||isNaN(sid0[1])
 ||isNaN(sid0[2]))
return 'eyeSideUp() error 04'

var len0;
len0=length1(eye0);
eye0=[eye0[0]/len0,eye0[1]/len0,
      eye0[2]/len0];
len0=length1(sid0);
sid0=[sid0[0]/len0,sid0[1]/len0,
      sid0[2]/len0];

var up0=v1Xv2(eye0,sid0); //a006151150
 up0=vec1(up0);
var sid1=v1Xv2(up0,eye0); //a006151153

var rotMat=[];
rotMat[0]=[]
rotMat[1]=[]
rotMat[2]=[]

for(var i=0;i<3;i++)
  {
rotMat[i][0]=eye0[i]
rotMat[i][1]=sid1[i]
rotMat[i][2]= up0[i]
  }

return rotMat; //a006151156
} //function eyeSideUp(arg1,arg2,arg3)






//a005182055
//from quaternion to matrix
//a005182102 copied from 
//function quat2rotMatrix1() 
//quat2rotMatrix1()  depend on
//html element.
//q2m(q,t) not depend on html element.
//
//Input q is quaternion, for example
//q=[0.8660254037844387,0.13363062095621217
//   0.26726124191242434,0.40089186286863654]
//q=[cos(θ),sin(θ)*v1,sin(θ)*v2,sin(θ)*v3]
//θ is HALF rotation angle in rad.
//t is return type
//if t=0, return string matrix
//otherwise return array matrix.
//
function q2m(q,t) //a005182050
{
if(typeof(q)!='object')
return 'q2m() error 1:typeof(q)!=object';
if(q.length!=4)
return 'q2m() error 2:q.length!=4';
var i,j,k;
if(typeof(q[0])=='string')
  {
var quat=[];
for(i=0;i<q.length;i++)
    {
quat[i]=parseFloat(q[i]);
if(isNaN(quat[i]))
return 'q2m() error 3:isNaN(quat[i]), q['+i+']='+q[i];
    }
q=quat;
  }
k=0;
for(i=0;i<q.length;i++)
k+=q[i]*q[i];
//if(Math.sqrt(Math.abs(k-1.0))>1.e-8)
//a005182159 del Math.sqrt()
if(Math.abs(k-1.0)>1.e-8)
  return 'q2m() error 4:abs(q)>1.e-8, '+Math.sqrt(Math.abs(k-1.0));

//above are filter, make sure
//code below is for real quaternion
//a005182116

var q0=q[0];
var q1=q[1];
var q2=q[2];
var q3=q[3]; //a005182202

//Jack B.Kuipers page 126 
//matrix is q*v*q_transpose
//from q*v*q_transpose get 
//9710181331 calculation result.
//
//9710181331
q11=bye09a(2*q0*q0-1+2*q1*q1);
q12=bye09a(2*q1*q2-2*q0*q3);
q13=bye09a(2*q1*q3+2*q0*q2);
q21=bye09a(2*q1*q2+2*q0*q3);
q22=bye09a(2*q0*q0-1+2*q2*q2);
q23=bye09a(2*q2*q3-2*q0*q1);
q31=bye09a(2*q1*q3-2*q0*q2);
q32=bye09a(2*q2*q3+2*q0*q1);
q33=bye09a(2*q0*q0-1+2*q3*q3);

if(t==0) //return string matrix
return '' //a003162057
+q11+', '+q12+', '+q13+'\n'
+q21+', '+q22+', '+q23+'\n'
+q31+', '+q32+', '+q33+'\n'

else //return array matrix
return [ //a005182100
[q11,q12,q13],
[q21,q22,q23],
[q31,q32,q33]]
} //function q2m(q,t) //a005182120







//9912201319
//function s2a() exist. expect 
//reverse function name as a2s()
//but a2s() does not exist
//reverse function name as mat2str(arg1)
//it is not a2s()
//now write a a2s()
//do not get trouble again.
//9912201321
function a2s(arg1)
{ //9912201318
return mat2str(arg1)
}















//9912201300 document here 
var docE='<pre>'
+'<a name=docE001>&lt;a name=docE001&gt;</a>\n'
+'Below is document for mat2str() .\n'
+'[mat2str] mean from matrix to string.\n'
+'2010-06-28-18-24\n'
+'if define a 2D matrix as below\n'
+'var leftMat= //9906281819\n'
+'[[11,12,13,14,15],\n'
+' [16,17,18,19,20],\n'
+' [21,22,23,24,25]];\n'
+'<a name=docE002>&lt;a name=docE002&gt;</a>\n'
+'if print this number array directly,\n'
+'document.write(leftMat)\n'
+'get\n'
+'11,12,13,14,15,16,17,18,19,20,21,22,23,24,25\n'
+'Structure of 2D disappear.\n'
+'<a name=docE003>&lt;a name=docE003&gt;</a>\n'
+'function mat2str(arg1) read number array\n'
+'[[11,12,13,14,15],\n'
+' [16,17,18,19,20],\n'
+' [21,22,23,24,25]];\n'
+'At the point change row, insert a newline.\n'
+'that is change "]," to a newline byte.\n'
+'In same row inbetween numbers, add space\n'
+'<a name=docE004>&lt;a name=docE004&gt;</a>\n'
+'output as following\n'
+' 11  12  13  14  15\n'
+' 16  17  18  19  20\n'
+' 21  22  23  24  25\n'
+'output above is better than output below.\n'
+'11,12,13,14,15,16,17,18,19,20,21,22,23,24,25\n'
+'<a name=docE005>&lt;a name=docE005&gt;</a>\n'
+'function mat2str(arg1) main purpose is to\n'
+'display matrix in easy to read form.\n'
+'2010-06-28-18-29\n'
+'\n'
+'</pre>'

//9906171542 write as in button code
//9906181038 change to function.

function mat2str(arg1)
{ 
var i,j,o='';

if(typeof arg1=='string')
return arg1; //input is already a string

if(typeof arg1=='object'
 &&typeof arg1[0]=='number'
  ) //input is a row data
  { 
for(i=0;i<arg1.length;i++)
o+=bye09v02(arg1[i])+'  ';
return o;
  }
//9906202301 use bye09v02()

//input is 2D matrix
for(i=0;i<arg1.length;i++)
  {
for(j=0;j<arg1[i].length;j++)
    {
o+=bye09v02(arg1[i][j])+'  ';//read one row data
    }
o+='\n'; //change row, change line
  }

return o; 

} //9906181048 done function mat2str(arg1)













/**
xy2xyArray(inxy) change from
[[
0,    1.963
0.068,1.868
0.134,1.715
]]
to
[[
MyLine1.x=[0,0.068,0.134]
MyLine1.y=[1.963,1.868,1.715]
MyLine1.VMLstroke="weight=1; color=black; dashstyle=solid;";
MyGraph.Plot(MyLine1);
]]
/**/
//function xy2xyArray(inxy)
//2011-06-14-17-27 rename
function XYGraphCode(inxy)
{ //9912121623

if(typeof(inxy)!='string')
return 'input is not string.'

if(inxy.length==0)
return 'input is empty.'

var wkStr0=''; //9912112330
var array1=''
var array2=''
var w1,w2,w3;
var i,j,k;

array1=inxy.split('\n');
array2=[];
for(j=0;j<array1.length;j++)
    { //9912112346
if(array1[j].length>0)
array2[j]=array1[j];
else break;
    }
for(j=0;j<array2.length;j++)
    { //9912112348
array2[j]=array2[j].split(',');
    }
for(j=0;j<array2.length;j++)
for(k=0;k<array2[j].length;k++)
    { //9912112351 use parseFloat() drop '\n'
array2[j][k]=parseFloat(array2[j][k]);
    }

var oStr4=''
+'MyLine1.x=['

for(j=0;j<array2.length;j++)
    { //9912112354
oStr4+=array2[j][0]
+(j<array2.length-1?',':'')
    }
oStr4+=']\n';

oStr4+=''
+'MyLine1.y=['
for(j=0;j<array2.length;j++)
    { //9912112355
oStr4+=array2[j][1]
+(j<array2.length-1?',':'')
    }
oStr4+=']\n'
;

if(typeof(QSTriangleclr)=='object'
 &&typeof(QSTrianglewth)=='object'
 &&typeof(QSTrianglesdd)=='object'
  ) //9912141511 use if()
oStr4+='' //9912121628
+'MyLine1.VMLstroke="weight='+QSTrianglewth.value
+'; color='+QSTriangleclr.value
+'; dashstyle='+QSTrianglesdd.value
+';";\n'
+'MyGraph.Plot(MyLine1);\n' //9912141513
else
oStr4+='' //9912121628
+'MyLine1.VMLstroke="weight=1'
+'; color=#000000'
+'; dashstyle=solid'
+';";\n'
+'MyGraph.Plot(MyLine1);\n' //9912121629

return oStr4; //9912121630
} //function xy2xyArray(inxy)
  //function XYGraphCode(inxy)











//function mat33det(arg1)
//find 3 by 3 matrix determinant value 
//input is string type 3 by 3 matrix
//
function mat33det(arg1)
{ //a003181101
var mat33;
mat33=s2aEZ(arg1);

if(mat33.length!=3)
return 'Matrix do not have three rows, now='+mat33.length;
if(mat33[0].length!=3)
return 'Matrix do not have three columns, now='+mat33[0].length;

var Mdet=0 //curve3d2.htm 9710181350
+mat33[0][0]*mat33[1][1]*mat33[2][2]
+mat33[0][1]*mat33[1][2]*mat33[2][0]
+mat33[0][2]*mat33[1][0]*mat33[2][1]
-mat33[0][2]*mat33[1][1]*mat33[2][0]
-mat33[0][1]*mat33[1][0]*mat33[2][2]
-mat33[0][0]*mat33[1][2]*mat33[2][1];

//a003182133 change next output 
if(!isNaN(Mdet))
return Mdet
else //a003181300 if() else
return 'error: 3 by 3 matrix must be numerical, can not be text.\n'

} //function mat33det(arg1) a003181118











/**
a003222311
function transposef(arg1,arg2) transpose matrix
arg1 is before transpose, original matrix 
for example
1,4,7,9,6,3
2,5,8,8,5,2
3,6,9,7,4,1

after transpose, the transposed matrix is
1,2,3
4,5,6
7,8,9
9,8,7
6,5,4
3,2,1

Above is numerical matrix
below is symbolic  matrix
 cos(ψ), sin(ψ), 0
-sin(ψ), cos(ψ), 0
   0,     0,     1
after transpose, change to
 cos(ψ),-sin(ψ),   0
 sin(ψ), cos(ψ),     0
 0, 0,     1

string type matrix is
'1,4,7,9,6,3
 2,5,8,8,5,2
 3,6,9,7,4,1'

string ARRAY type matrix is
[['1','4','7','9','6','3'],
 ['2','5','8','8','5','2'],
 ['3','6','9','7','4','1']]
a003222327

/**/
//function transposef(arg1)
function transpoMat(arg1)
{ //a003171420 transposef()
  //a006182140 transpoMat()

//change string to array of string
function slic0(q){if(typeof(q)!='string')return q;if(q.indexOf(',')>0)q=q.split(',');else{while(q.indexOf('  ')>=0)q.replace('  ',' ');var i=0;var j=q.length-1;if(q.charAt(i)==' ')i++;if(q.charAt(j)==' ')j--;q=q.substring(i,j+1);q=q.split(' ');}return q;}//a006181845

//delete line end change line bytes
function cutEnd(q){var i=q.length-1;while(q.charCodeAt(i)<32)i--;q=q.substring(0,i+1); return q;}  //a006190612

var returnStr=0; //a006182141
var inp1=arg1;
var i,j,k;
var ans1=[];

if(typeof(arg1)=='string')
  { //a006182037 if()
returnStr=1;
inp1=cutEnd(inp1);
inp1=inp1.split('\n');

for(i=0;i<inp1.length;i++)
inp1[i]=slic0(cutEnd(inp1[i])); 
//a006182155 slic0;a006190614cutEnd
  }
else
  {
inp1=arg1; //a006182038
  }

if(typeof(inp1)=='object' //a003231218
 &&(typeof(inp1[0])=='number'
  ||typeof(inp1[0])=='string'))
  {
if(returnStr==1)
    {
var ans0=''; //a003231227
for(i=0;i<inp1.length;i++)
//next line add a newline byte
//change row data to column data
ans0+=inp1[i]+'\n'; //a003231228

return ans0; //a003231229
    }
else
    {
for(i=0;i<inp1.length;i++)
      {
ans1[i]=[];
ans1[i][0]=inp1[i]
      }
return ans1; //a006182200
    }
  } //if(typeof(inp1)=='object' 單行向量

//above  one row to  one column
//below many row to many column

for(j=0;j<inp1[0].length;j++)
ans1[j]=''; //a003171446

for(i=0;i<inp1.length;i++)
  {
for(j=0;j<inp1[i].length;j++)
    {
k=parseFloat(inp1[i][j]);//a006191216
if(!isNaN(k))ans1[j]+=k  // +' # '
else
ans1[j]+=inp1[i][j] //transpose here at j

ans1[j]+=(i<inp1.length-1?',':'\n') 
//above change number to string
//at a006182211 recover to number
    }
  } 

var ans2; //a003171452

if(returnStr==1)
  {
ans2='';
for(i=0;i<ans1.length;i++)
ans2+=ans1[i];

return ans2; //a003171456
  } 

for(i=0;i<ans1.length;i++)
  {
//drop line end newline byte
ans1[i]=ans1[i].substring(0,ans1[i].length-1)
ans1[i]=slic0(ans1[i])
  }

//change string to number
//if know input is number array
//no string number ['1.23'], no
//symbol string ['sin(1.2)']
//then not need parseFloat() 
//If input has string number 
//parseFloat() is just right.
//If input has symbol string
//k=parseFloat('sin(1.2)') get 
//isNaN(k)==true, then use 
//original 'sin(1.2)' forget NaN
//This design allow symbol string
//matrix be transposed.
//a006182358
//
for(i=0;i<ans1.length;i++)
for(j=0;j<ans1[i].length;j++)
  {
//input 'sin(1.2)' void parseFloat() 
k=parseFloat(ans1[i][j]);//a006182211
if(!isNaN(k))ans1[i][j]=k;  
  } 

return ans1; //a006182218

} //function transpoMat(arg1) 











//a004092050 document start
//rotateShift(arg1,arg2,arg3,arg4)
//arg1 is rotation matrix , for example
//  0,1,0
// -1,0,0
//  0,0,1
//
//arg2 is 3D point x,y,z coordinates
//  1,2,3 // (x,y,z)=(1,2,3)
//  4,5,6
//
//arg3 is rotation pivot point
//for example (x,y,z)= 1,2,3 
//
//arg4 is 3D parallel move increment
//value, for example (x,y,z)= 4 5 6 
//a004092057 document stop
//
//2011-03-17-18-42 change button code
//to function 
//
function rotateShift(arg1,
      arg2,arg3,arg4)
{
var ou1=''; //a003171102
var mat1=s2aEZ(arg1);
var mat2=s2aEZ(arg2);
var pivot0=s2aEZ(arg3);
var xyzgap=s2aEZ(arg4);

if(pivot0.length!=3) //a003171114
return '[pivot rotate parallel move] button error01\n'
+'Pivot point coordinate must be three numbers,\n'
+'now read '+pivot0

if(xyzgap.length!=3) //a003171116
return '[pivot rotate parallel move] button error02\n'
+'3D parallel move increment value must be three numbers,\n'
+'now read '+xyzgap

var xyzinc=[];
var mat2a=[];
var matmat=[];
var i,j,k;
for(i=0;i<pivot0.length;i++) //a003171110
xyzinc[i]=pivot0[i]+xyzgap[i];

if(typeof(mat2[0])=='number')
{k=mat2;mat2=[];mat2[0]=k;k='';} //a004092108

//rotate just one point (input row vector)
//and rotate many points (input matrix)
//merge to one code. change one variable 
//vector to two variable vector
for(i=0;i<mat2.length;i++)
    {
for(j=0;j<mat2[i].length;j++)
      {
//a003171145 add [i]
//a004050013 add "-"
mat2a[j]=+mat2[i][j]-pivot0[j]; 
      }
matmat=matMatNumberV01(mat1,mat2a); //a003171142

for(j=0;j<matmat.length;j++)
      {
//a004042326 add [0]
ou1+=(matmat[j][0]+xyzinc[j])
+(j<matmat.length-1?',':'\n');
      }
    }

return ou1; //a004042304
} //function rotateShift(...)












//a003161606 copy function matMatNumberV01()
//from sphere3d.htm to rotate3c.htm
//
var docD='<pre>'
+'<a name=docD001>&lt;a name=docD001&gt;</a>\n'
+'Next is function matMatNumberV01()\n'
+'explanation.\n'
+'2010-06-28-12-09 start\n'
+'function matMatNumberV01(arg1,arg2)\n'
+'Initially design function as matrix multiply\n'
+'vector, later change function to matrix multiply\n'
+'matrix ans still use same function name\n'
+'\n'
+'<a name=docD002>&lt;a name=docD002&gt;</a>\n'
+'function matMatNumberV01(arg1,arg2) and\n'
+'function matMatStr(arg1,arg2,arg3)  both run\n'
+'left matrix arg1 multiply right matrix arg2\n'
+'matMatNumberV01() input 2D number array,\n'
+'matMatStr()       input 2D string array.\n'
+'<a name=docD003>&lt;a name=docD003&gt;</a>\n'
+'For example, 2D matrix\n'
+' 11 12 13 14 15  ---eqn.001\n'
+' 16 17 18 19 20\n'
+' 21 22 23 24 25\n'
+'2D number array expression is\n'
+'<a name=docD004>&lt;a name=docD004&gt;</a>\n'
+'var leftMat=\n'
+'[[11,12,13,14,15], ---eqn.002\n'
+' [16,17,18,19,20],\n'
+' [21,22,23,24,25]];\n'
+'If express with string, it is\n'
+'\'11 12 13 14 15 16 17 18 19 20 21 22 23 24 25\' ---eqn.003\n'
+'<a name=docD005>&lt;a name=docD005&gt;</a>\n'
+'if user define matrix from input Box,\n'
+'All input Box value can be string only.\n'
+'Even if copy eqn.002 to box, it is still\n'
+'a string, not a number array.\n'
+'Function s2a(QHboxb11.value) change \n'
+'eqn.001 (string) to eqn.002 (number array)\n'
+'s2a() see newline byte as change row.\n'
+'ifcopy eqn.003 string to input Box, \n'
+'no newline, function s2a() can not work.\n'
+'<a name=docD006>&lt;a name=docD006&gt;</a>\n'
+'if change eqn.003 to next eqn.004\n'
+'\'11 12 13 14 15;16 17 18 19 20;21 22 23 24 25\' ---eqn.docD004\n'
+'although no newline, but has \';\' ,function s2a()\n'
+'can work. Because s2a() change \';\' to newline.\n'
+'//In rotate3d.htm please use newline, not use \';\' a004211417\n'
+'<a name=docD007>&lt;a name=docD007&gt;</a>\n'
+'matMatNumberV01() read data from eqn.002 \n'
+'matrix row number is arg1.length\n'
+'matrix column number is arg1[0].length\n'
+'use two variable to get matrix element\n'
+'value, for example arg1[i][j]\n'
+'<a name=docD008>&lt;a name=docD008&gt;</a>\n'
+'Above is matMatNumberV01() \n'
+'Below is matMatStr()\n'
+'if left matrix is 2 row and 3 column\n'
+'1 2 3 ---eqn.005\n'
+'4 5 6\n'
+'if right matrix is 3 row and 2 column\n'
+' 7  8  ---eqn.006\n'
+' 9 10\n'
+'11 12\n'
+'<a name=docD009>&lt;a name=docD009&gt;</a>\n'
+'matMatStr() read input\n'
+'arg1= \'1 2 3 4 5 6\' and\n'
+'arg2= \' 7  8  9 10 11 12\'\n'
+'in this case need third argument\n'
+'arg3= 3\n'
+'arg3 tell matMatStr() left/right two \n'
+'matrices have common dimension 3, then\n'
+'<a name=docD010>&lt;a name=docD010&gt;</a>\n'
+'matMatStr() can build matrix\n'
+'eqn.005 and eqn.006\n'
+'get correct answer.\n'
+'\n'
+'matMatNumberV01() and matMatStr() are\n'
+'two very different function, they do \n'
+'the same job.\n'
+'2010-06-28-12-30 stop\n'
+'\n'
+'<a name=docD011>&lt;a name=docD011&gt;</a>\n'
+'2010-06-16-18-01\n'
+'arg1 define left matrix\n'
+'arg2 define right matrix\n'
+'2010-06-16-21-25 allow row not equal column\n'
+'if matrix is\n'
+'[1  2  3  4]  ---eqn.007\n'
+'[5  6  7  8]\n'
+'[9 10 11 12]\n'
+'input as number array, otherwise not compatible.\n'
+'<a name=docD012>&lt;a name=docD012&gt;</a>\n'
+'&lt;textarea&gt; web page Box value can be string only,\n'
+'change a string\n'
+'\'1 2 3 4 5 6 7 8 9 10 11 12\'\n'
+'to number array, need function to do the job.\n'
+'function s2a(arg1,arg2,arg3)\n'
+'input string, output number array.\n'
+'<a name=docD013>&lt;a name=docD013&gt;</a>\n'
+'if arg2 define n element vector (not matrix),\n'
+'then left matrix arg1 must have m*n elements.\n'
+'\n'
+'function matMatNumberV01() need\n'
+'function readdata()\n'
+'2010-06-16-21-16\n'
+'\n'
+'//matMatNumberV01(boxData.value) cause error\n'
+'//matMatNumberV01(s2a(boxData.value)) get correct output .\n'
+'//2010-06-28-19-12\n' //9912061221 add
+'</pre>'
;







//9912061040 from jsmajor2.htm
//copy matMatNumberV01() to sphere3c.htm
//and delete "var QGbx18=0;"
//
//function matMatNumberV01(arg1,arg2)
//a003221752 change arg1,arg2 to inMat1,inMat2
//
function matMatNumberV01(inMat1,inMat2)
{ 
//9906171052 version 1 matMVecf_ver1()
//9906221511 version 2 matMVecf_ver2()
//9912061647 version 3 matMVecf_ver3()
//a003221640 version 4 matMVecf_ver4()
//Matrix Multiply with Vector
//Left matrix multiply with right vector
//Left matrix multiply with right matrix
//input must be number array, can not be string
//call in between functions, it is better
//use number array. On the other hand,
//read from web page box, it is string
//only. Box value can not be array
//9906161902 begin matMVecStr()
//9906162151 done  matMVecStr()

if((typeof inMat1)=='string'
 ||(typeof inMat2)=='string'
  ) //9912061649
  {
return ['function matMatNumberV01() error01\n',
        'input can not be string.'];
  }

//a004211643 use if()
if(typeof(QBspanAlert02)=='object')
QBspanAlert02.innerHTML=''; //a004101713

status=''; //a004101151

var ans2=[];
var i,j,k;

//next is code body 1
// inMat1 (vector) multiply
// with inMat2 (pure number)

if(typeof(inMat1)=='object'
 &&typeof(inMat1[0])=='number'
 &&typeof(inMat2)=='object'
 &&typeof(inMat2[0])=='number'
  ) //a004092147
  {

if(inMat1.length!=inMat2.length) //a004101126 if()
for(i=0;i<inMat1.length;i++)
    {
ans2[i]=inMat1[i]*inMat2[0]; //a004092141
    }
else
    {
ans2=0; //a004101127
for(i=0;i<inMat1.length;i++)
ans2+=inMat1[i]*inMat2[i]; //a004101128
    }
return  ans2;
  }


var sum2=0;

//a003221705 below move out from
//code body 3 for public use
var rowDim1, colDim1; //9906162256
var rowDim2, colDim2; //9906162257
var i,j,k;

if(typeof(inMat1)=='number')
{rowDim1=colDim1=1;} //a003221714
else
if(typeof(inMat1)=='object'
 &&typeof(inMat1[0])=='number'
  )
{rowDim1=1;colDim1=inMat1.length;} //a003221803
else
  {
rowDim1=inMat1.length;  //9906162258
colDim1=inMat1[0].length;
  }

if(typeof(inMat2)=='number')
{rowDim2=colDim2=1;} //a003221716
else
if(typeof(inMat2)=='object'
 &&typeof(inMat2[0])=='number'
  )
  {
colDim2=1;rowDim2=inMat2.length; //a003221736

var tmp=[]
for(i=0;i<inMat2.length;i++)
{tmp[i]=[];tmp[i][0]=inMat2[i];} //a003221746
inMat2=tmp; //a003221740
  }
else
  {
rowDim2=inMat2.length;  //9906162259
colDim2=inMat2[0].length;
  }

//a004211715 why error
//copy from backup no error file
//then paste recent translation
//(may contain error) to no error file
//a004211751 why not find error?
//up to here it is OK



if(colDim1!=rowDim2)
    {
return ['function matMatNumberV01() error02\n'
+'Left matrix and right matrix dimension are not compatible.\n'
+'Left matrix column number must be the same as right matrix\n'
+'row number. Now two matrices dimension are:\n'
+' Left matrix rowDim1='+rowDim1+'\n'
+' Left matrix colDim1='+colDim1+'\n'
+'Right matrix rowDim2='+rowDim2+'\n'
+'Right matrix colDim2='+colDim2+'\n'


+'Above four numbers, middle two must be the same.\n'
+'If different, can not carry out matrix multiplication.\n'
+'9912061216\n','']
    }
//a003221705 move out above from
//code body 3



//below is code body 3
// left matrix inMat1 multiply
//right matrix inMat2.
if((typeof inMat1)=='object'
 &&(typeof inMat1[0])=='object'
 &&(typeof inMat1[0][0])=='number'
 &&(typeof inMat2)=='object'
 &&(typeof inMat2[0])=='object'
 &&(typeof inMat2[0][0])=='number'
  ) //9912061702
  {
//next three for() loops do the
//matrix multiplication
for(i=0;i<rowDim1;i++)
    {
ans2[i]=[]; //9912061715
for(k=0;k<colDim2;k++)
      {
sum2=0;
for(j=0;j<rowDim2;j++)
        {
sum2+=inMat1[i][j]*inMat2[j][k]
        }
//ans2[i][k]=sum2;
ans2[i][k]=bye09v02(sum2); //9906231612 use bye09v02()
//bye09v02() change 1.99999999999 to 2
//bye09v02() change 1.00000000002 to 1
      }
    } //for(i=0;i<rowDim1;i++)

return  ans2;
  }

//below is code body 5
// left row vector inMat1 multiply
//right matrix inMat2.
if((typeof inMat1)=='object'
 &&(typeof inMat1[0])=='number'
 &&(typeof inMat2)=='object'
 &&(typeof inMat2[0])=='object'
 &&(typeof inMat2[0][0])=='number'
  ) //a003221812
  {

ans2=[];
for(i=0;i<colDim2;i++)
    {
sum2=0;
for(j=0;j<rowDim2;j++)
      {
sum2+=inMat1[j]*inMat2[j][i]; //a003222054
      } //for(j=0;j<colDim2;j++)
ans2[i]=sum2;
    } //for(i=0;i<rowDim2;i++)

return  ans2;  //9912061833
  } //if(left row vector multiply right matrix)


return ['function matMatNumberV01() error09\n'
+'Did not find workable input.\n','']; //9912061716

} //function matMatNumberV01()
//9906171350 done function matMVecf_ver1(arg1,arg2)
//9906231612 done function matMVecf_ver2(arg1,arg2)
//9912061841 done function matMVecf_ver3(arg1,arg2)
//9912201259 document here











//a003310938 rename function name
//to run3Dto2D(inp7)
//
//was test3Dto2D(inp7)
//now run3Dto2D(inp7)
//main change is return form
//a003310941 change to next
//return [xy2xyArray(oup7), oup7]
//return [XYGraphCode(oup7), oup7]
//
//minor change is from inp7 to inp0
//a003310932 record
//
//a005171550 add eyexyz,upxyz
//so that six html element QSboxEyeX
//to QSbox_UpZ can be deleted from
//function run3Dto2D()
//
function run3Dto2D(inp0,
eyexyz,upxyz  //a005171550
)
{ //9912121604

if(inp0.length==0)
return 'Input string is empty; run3Dto2D(inp0)'

if(typeof(eyexyz)!='object')
var eyexyz=[3,2,1]; //a005171558

if(typeof(upxyz)!='object')
var upxyz=[0,0,1]; //a005171559

//9912111852
var parallelOrPerspective=1; //parallel (cylinder)

if(typeof(QAchkbxID05)=='object') //a005171553 typeof
if(QAchkbxID05.checked)
parallelOrPerspective=0; //Perspective (cone)

var scrPercent=50;
if(parallelOrPerspective==0)
if(typeof(QAboxScrP)=='object') //a005171554 typeof
scrPercent=parseFloat(QAboxScrP.value); //9912111931

var oup7= dim3to2b(inp0,  //9912121610
eyexyz, //eye x,y,z location
upxyz,  //screen up direction x,y,z
scrPercent ,  //60%  (0,0,0) < screen < eye ; scrPercent
parallelOrPerspective //1=parallel, non-1=Perspective
);

//return xy2xyArray(oup7)+'\n\n/**\n'+oup7+'\n/**/\n'
//a003310941 change to next
//return [xy2xyArray(oup7), oup7]
//xy2xyArray() rename to next
return [XYGraphCode(oup7), oup7]

} //function run3Dto2D(inp0) 9912121632












//dim3to2b is dim 3 to dim 2, version b
//function dim3to2() access html elements
//function dim3to2b() read input arguments
//
//if arg1 is not string, change to string
//if(typeof(inStr0)=='object')
//inStr0=inStr0+''; //a002261436
//
//
function dim3to2b(
//inStr0 require one line one x,y,z data set
inStr0,     //arg1, 
eyexyz,     //arg2, //eye x,y,z location
scrUPxyz,   //arg3, //screen up direction x,y,z
scrPercent, //arg4, //(0,0,0) < screen < eye
//parallelOrPerspective==1 eye at infinity, 
//else eye is near 9912101121
parallelOrPerspective  //arg5
                 ) //9912092025
//function dim3to2b() return ouStr2
//ouStr2 is 2d x,y data in string form
{
var i,j,k;
var err0=0;

//if arg1 is not string, change to string
if(typeof(inStr0)=='object')
  {
if(typeof(inStr0[0])=='number')
inStr0=inStr0+''; //a002261436
else 
if(typeof(inStr0[0])=='object')
    {
if(typeof(inStr0[0][0])=='number')
      {
k='';
for(i=0;i<inStr0.length;i++)
k+=inStr0[i]+'\n';
inStr0=k; //a002261503
      }
else err0=1;
    }
else err0=1; //a002261456

if(err0==1)
return 'First argument must be number in string form.\n'
+'or one dimensional number array example [1,2,3]\n'
+'or two dimensional number array example next line\n'
+'[[1,2,3],[4,5,6],[7,8,9]] not require integer.\n'
+'but must be three number one set for x,y,z.\n'
+'Three dim. number array or higher not supported.\n'
+'Now first argument is\n['+inStr0+']\n';
  } //if(typeof(inStr0)=='object')

i=inStr0.lastIndexOf('\n')
if(i<inStr0.length)
inStr0=inStr0+'\n'; //9912101844

 var e0,e1,e2,e3;
 var tiny0=1.e-6; //9712172112

 var ouStr2='';
e0=inStr0.indexOf('\n',0); //9712281810

if(e0<0)
  {
ouStr2= //9912092033 change to 'ouStr2='
"Program require one point x,y,z data one line."; //9712281820
return ouStr2;
  }
e0=inStr0.indexOf(',',0); //9712281804



if(e0<0)
  {
e0=inStr0.indexOf('\t',0); //9712281809
if(e0>0)
    {
inStr0=inStr0.replace(/\t/g,','); //9712281811
    }
  else
    {
for(e1=0;e1<inStr0.length;e1++)
      {
if(inStr0.charAt(e1)!=' ')
  ouStr2+=inStr0.charAt(e1);
else
if(inStr0.charAt(e1-1)!=' ')
  ouStr2+=',';
else
  ouStr2+=inStr0.charAt(e1);
      }
inStr0=ouStr2; //9712281832 make up ','
    }
ouStr2=''; //9712281835
  }


if(arguments.length>2)  //9912101523 use if()
  {
eyexyz[0]=parseFloat(eyexyz[0]); //9712161851
eyexyz[1]=parseFloat(eyexyz[1]);
eyexyz[2]=parseFloat(eyexyz[2]);
  }
else
  {
eyexyz=[1,0,0];   //9912101538
  }



if(isNaN(eyexyz[0]))
  {
eyexyz[0]=1; 
  }
if(isNaN(eyexyz[1]))
  {
eyexyz[1]=0  
  }
if(isNaN(eyexyz[2])) //9712191318
  {
eyexyz[2]=0; 
  }

  eyedist=Math.sqrt(eyexyz[0]*eyexyz[0]
+eyexyz[1]*eyexyz[1]+eyexyz[2]*eyexyz[2]);
if(eyedist<tiny0)
  {
eyexyz[0]=1; //9912100945 
eyexyz[1]=   //use default 
eyexyz[2]=0; //eyexyz=[1,0,0]
  }

//9912111835 add '/100'
scrPercent=parseFloat(scrPercent)/100;

if(isNaN(scrPercent) //9712191320
  ||scrPercent==''   //9712281943
  )
  {
scrPercent=0.5; //9912100937 use default
  }
var scrdist=eyedist*scrPercent;

var screenXY=new Array(2); //9712161601 this is answer

if(arguments.length>3)  //9912101537 use if()
  {
scrUPxyz[0]=parseFloat(scrUPxyz[0]);
scrUPxyz[1]=parseFloat(scrUPxyz[1]);
scrUPxyz[2]=parseFloat(scrUPxyz[2]); //9712161853
  }
else
  {
scrUPxyz=[0,0,1]; //9912101535
  }

if(isNaN(scrUPxyz[0])
 ||isNaN(scrUPxyz[1])
 ||isNaN(scrUPxyz[2])
  )
  {
scrUPxyz[0]=   //9912100942 
scrUPxyz[1]=0; //use default
scrUPxyz[2]=1; //scrUPxyz=[0,0,1]
  }

if(isNaN(parallelOrPerspective))
  { //9912101749
parallelOrPerspective=1; 
  }

//based on eye coordinate eyexyz
//and screen-up coordinate scrUPxyz
//find new screen-up coordinate wkxyz
wkxyz=eyeXup(eyexyz,scrUPxyz); //9712311039
//scrUPxyz is user estimate value
//wkxyz is exact 90 degree vector

e3=Math.sqrt(wkxyz[0]*wkxyz[0]+wkxyz[1]*wkxyz[1]
+wkxyz[2]*wkxyz[2]);
scrUPxyz[0]=wkxyz[0]/e3; //new screen-up 
scrUPxyz[1]=wkxyz[1]/e3; //coordinate 
scrUPxyz[2]=wkxyz[2]/e3; //scrUPxyz
//9712171839 hope right
//9712171924 three vectors co-plane
e0=eyexyz[0]*scrUPxyz[1]*wkxyz[2]
+scrUPxyz[0]*wkxyz[1]*eyexyz[2]
+wkxyz[0]*eyexyz[1]*scrUPxyz[2]
-eyexyz[0]*scrUPxyz[2]*wkxyz[1]
-scrUPxyz[0]*eyexyz[1]*wkxyz[2]
-wkxyz[0]*scrUPxyz[1]*eyexyz[2];
//e0 should be zero 9712171924

//corrected scrUPxyz should be
//perpendicular to eyexyz 9712171950
e1= //e1 should be zero 9712171951
 wkxyz[0]*eyexyz[0]
+wkxyz[1]*eyexyz[1]
+wkxyz[2]*eyexyz[2]
  var ouStr3=
 'eye vector and screen-up vector both begin at (0,0,0) \n'
+'User input two vectors may not be perpendicular to each other.\n'
+'Program correct screen-up vector make it perpendicular to eye\n'
+'vector. Eye vector, user-screen-up vector, corrected-screen-up\n'
+'vector, three of them co-plane. their volume is zero.9712171842\n'
+'\nEye eyexyz[0]='+eyexyz[0]
+'\nEye eyexyz[1]='+eyexyz[1]
+'\nEye eyexyz[2]='+eyexyz[2]

+'\nUser scrUPxyz[0]='+scrUPxyz[0]
+'\nUser scrUPxyz[1]='+scrUPxyz[1]
+'\nUser scrUPxyz[2]='+scrUPxyz[2]

+'\nCorrected scrUPxyz[0]='+scrUPxyz[0]
+'\nCorrected scrUPxyz[1]='+scrUPxyz[1]
+'\nCorrected scrUPxyz[2]='+scrUPxyz[2]

+'\nAbove three vectors co-plane, check it below\n'
+e0
+'\nCorrected scrUPxyz should be perpendicular to eyexyz \n'
+'their dot product should be zero, check below.\n'
+e1+'\n'  //9712171925
+'a006081903 output above to axs2mat2.htm:QBboxc16'

//9912101555 include "rotxyz=[[],,] line
 var rotxyz=[[0,0,0],[0,0,0],[0,0,0]]; //9712261600use

rotxyz[0][0]=eyexyz[0]/eyedist;
rotxyz[1][0]=eyexyz[1]/eyedist;
rotxyz[2][0]=eyexyz[2]/eyedist;
rotxyz[0][2]=scrUPxyz[0];
rotxyz[1][2]=scrUPxyz[1];
rotxyz[2][2]=scrUPxyz[2];

/**
9712291017 
y=z cross product x  9712261533
 y0  y1  y2
 z0  z1  z2
 x0  x1  x2

y0=z1*x2-z2*x1
y1=z2*x0-z0*x2
y2=z0*x1-z1*x0

rotxyz[0][y]=rotxyz[1][z]*rotxyz[2][x]-rotxyz[2][z]*rotxyz[1][x];
rotxyz[1][y]=rotxyz[2][z]*rotxyz[0][x]-rotxyz[0][z]*rotxyz[2][x];
rotxyz[2][y]=rotxyz[0][z]*rotxyz[1][x]-rotxyz[1][z]*rotxyz[0][x];
above and below same meaning 9712291032
/**/
rotxyz[0][1]=rotxyz[1][2]*rotxyz[2][0]-rotxyz[2][2]*rotxyz[1][0];
rotxyz[1][1]=rotxyz[2][2]*rotxyz[0][0]-rotxyz[0][2]*rotxyz[2][0];
rotxyz[2][1]=rotxyz[0][2]*rotxyz[1][0]-rotxyz[1][2]*rotxyz[0][0];
//9712291026

//9912112023 here is right location
var transpRot=[[0,0,0],[0,0,0],[0,0,0]]; //9712261950use

//define rotation matrix first then
//define transpose matrix
for(e0=0;e0<3;e0++) //9712291033
for(e1=0;e1<3;e1++)
transpRot[e0][e1]=rotxyz[e1][e0];

var matdet
=rotxyz[0][0]*rotxyz[1][1]*rotxyz[2][2]
+rotxyz[0][1]*rotxyz[1][2]*rotxyz[2][0]
+rotxyz[2][1]*rotxyz[0][2]*rotxyz[1][0]
-rotxyz[1][1]*rotxyz[0][2]*rotxyz[2][0]
-rotxyz[0][1]*rotxyz[2][2]*rotxyz[1][0]
-rotxyz[2][1]*rotxyz[1][2]*rotxyz[0][0];
//9712261545

//9912101610 include iter0
  var line0=''; //9712152154
  var pointxyz; //9712152209
  var isNaNX=isNaNY=0; //9712191248
  var iter0=0; //9712191256
  var rangeX0,rangeX1;
  var rangeY0,rangeY1; //9712191423
  var maxDist=minDist=0; //9712191740
  var p2IDistance; //9712191741

for(e1=0;e1<inStr0.length;e1++)
  { //9712152045 delete comment '//' success
if(inStr0.charAt(e1)=='/'
 &&inStr0.charAt(e1+1)=='/')
    { //input x,y,z data allow comment '//'
while(inStr0.charAt(e1)!='\n')e1++;
continue; //9712152152 this line correct
    }

  if(inStr0.charAt(e1)!='\n')
    {
  line0+=inStr0.charAt(e1);
  continue; //9712152156
    }
  else
    { //9712152158

//next call change space coord. x,y,z
//to 2D coord. x,y
ouStr0=//output is point-on-curve screen coord. x,y
xyz2xy( //9712241316
line0 //input is point-on-curve space coord. x,y,z
, transpRot //9912101059 add
, parallelOrPerspective //9912101120 add
, scrPercent //9912111743 add
, eyexyz //9912112126 add
);

line0=''; //already read, delete line0 9712241440
wkxyz=ouStr0.split(",");  //9712241322
screenXY[0]=parseFloat(wkxyz[0]); //2D x
screenXY[1]=parseFloat(wkxyz[1]); //2D y
p2IDistance=parseFloat(wkxyz[2]); //(x,y) to eye
if(isNaN(screenXY[0]))//Not a Number NaN skip
{isNaNX++; continue;} //sum how many skipped xyString
if(isNaN(screenXY[1]))
{isNaNY++; continue;} //9712191052
iter0++; //9712191257

ouStr2+=dotPrecision(screenXY[0])
   +','+dotPrecision(screenXY[1])
   +'\n';  //9912101758

    }   //if(inStr0.charAt(e1)!='\n') else

  }     //for(e1=0;e1<inStr0.length;e1++)

if(typeof(QBboxc16)=='object')
QBboxc16.value=ouStr3; //a006081903

return ouStr2;     //9912101150


} //function dim3to2b()
//9912201726
//function dim3to2b() return ouStr2;
//then what happen to ouStr3 ?
//LiuHH guess in 2008-12-28 write first
//version period, need debug and build
//ouStr3. Now decide not delete ouStr3
//future may need this verification code.
//9912201736
//a006081903 output ouStr3 to 
//axs2mat2.htm:QBboxc16















//3D coord. xyz change to 2D coord xy
function xyz2xy( //9712241313
line0,
transpRot, //9912101100 add transpRot is [][] array
//parallelOrPerspective=1 is parallel projection
//parallelOrPerspective=1 equal to eye at infinity
//parallelOrPerspective NOT=1 is eye at near
parallelOrPerspective //9912101114
, scrPercent //9912111744 add
, eyexyz //9912112128 add
)
{
//9712231553 3d2d begin [=]
var pointxyz=line0.split(","); 
//line0 data transfer to pointxyz
  line0=''; //then, delete line0

var scrCenter=new Array(3); //9712241419
 var tiny0=1.e-6; //9712172112

scrCenter[0]=scrPercent*eyexyz[0]
scrCenter[1]=scrPercent*eyexyz[1]
scrCenter[2]=scrPercent*eyexyz[2] //9712160912
pointxyz[0]=parseFloat(pointxyz[0]);
pointxyz[1]=parseFloat(pointxyz[1]);
pointxyz[2]=parseFloat(pointxyz[2]); //9712251457
if(Math.abs(eyexyz[0])<tiny0 //9712251732
 &&Math.abs(eyexyz[1])<tiny0
 &&Math.abs(eyexyz[2])<tiny0)//eye on [0,0,0] !?
  {
return 'eye is NOT allowed to stay on [0,0,0] !!'
  }

var matVecOut=[0,0,0]; //9712262014

//9912101111 calculate directly; not call matMulVec2()
matVecOut[0]= //9712262020 correction
 transpRot[0][0]*pointxyz[0]
+transpRot[0][1]*pointxyz[1]
+transpRot[0][2]*pointxyz[2];
matVecOut[1]=
 transpRot[1][0]*pointxyz[0]
+transpRot[1][1]*pointxyz[1]
+transpRot[1][2]*pointxyz[2];
matVecOut[2]=
 transpRot[2][0]*pointxyz[0]
+transpRot[2][1]*pointxyz[1]
+transpRot[2][2]*pointxyz[2];

var projRatio=(eyedist-scrPercent*eyedist)
    /(eyedist-matVecOut[0]); //9712271249

var ptxyzDist=Math.sqrt( //9712271558
 (eyexyz[0]-pointxyz[0])*(eyexyz[0]-pointxyz[0])
+(eyexyz[1]-pointxyz[1])*(eyexyz[1]-pointxyz[1])
+(eyexyz[2]-pointxyz[2])*(eyexyz[2]-pointxyz[2])
          );

//9912101117 change "chkbxID05" to next
if(parallelOrPerspective==1) 
xyString= //9712271559
// (matVecOut[1]*1)+', ' //parallel projection
//+(matVecOut[2]*1)+', ' //set projRatio to 1
//a001081422 drop '*1' save two calculations
 (matVecOut[1])+', ' // *1*1 key string
+(matVecOut[2])+', ' // for search
+ptxyzDist;
else
xyString= //9712271242 non-parallel projection
 (matVecOut[1]*projRatio)+', ' //projRatio is
+(matVecOut[2]*projRatio)+', ' //not equal to 1
+ptxyzDist;
//9712271252
//9712231552 3d2d output here [=]
return xyString; //9712241315
} //9712241456 end of function xyz2xy()













function eyeXup(eye0,up0) //9712311005
{
  var r0,r1;
 var tiny0=1.e-6; //9712172112
  var wk0=new Array(3);
  var wk1=new Array(3);
  wk2=[0,0,0];
//9712311024 eye vector cross screen-up vector
wk0=v1Xv2(eye0,up0); //result is wk0 vector
if(length1(wk0)<tiny0)return wk2; //wk0=0 vector? not allow
wk1=v1Xv2(wk0,eye0); //wk0 cross eye vector 
r0=length1(wk1);     //get corrected screen-up vector
if(r0<tiny0)return wk2;
wk1[0]=wk1[0]/r0; //corrected screen-up vector
wk1[1]=wk1[1]/r0; //length change to 1
wk1[2]=wk1[2]/r0;
return wk1; //return unit length corrected screen-up vector
} //9712311040 function eyeXup(eye0,up0)









//a006090848 to a006090920 
//was in-button code
//because use it three times
//change to function.
//
//a006090940 function phirho2xyz(inStr0)
//input is spherical coordinate φ,θ,ρ 
//output Cartesian coordinate x,y,z 
// φ polar angle phi φ //north pole φ=0 
// θ azimuthal angle theta θ //Greenwich θ=0 
// ρ radius distance rho ρ //earth center ρ=0 
//
//方位角	azimuthal angle	a007092016
//
//
function phirho2xyz(inStr0
 ,angleType //a006171559
  )
{ //a006090940
var q1,q2,x,y,z; //a006090848
q1=inStr0;
if(q1.indexOf(',')>0){q2=q1.split(',');}
else {while(q1.indexOf('  ')>=0)
q1=q1.replace(/  /g,' ');q2=q1.split(' ');}
if(q2.length==2)q2=[q2[0],q2[1],1];
else if(q2.length<2)q2=[0,0,1];
var phi=q2[0],tha=q2[1],rho=q2[2]; 
with(Math)
  {//a006090855
phi=phi*PI/180;
tha=tha*PI/180;

//緯度	latitude
//經度	longitude

if(angleType==1)
  { //change latitude to polar angle
phi=-phi+PI/2
  } //a006171603
x=dotPrecision(bye09v02(rho*sin(phi)*cos(tha)),5);
y=dotPrecision(bye09v02(rho*sin(phi)*sin(tha)),5);
z=dotPrecision(bye09v02(rho*cos(phi)),5);
  } //with(Math)

return x+','+y+','+z

} //function phirho2xyz(inStr0)
//a006090945












//a003102130 include cleanData(arg1)
//to tute0054.htm
//
function cleanData(arg1)
{ //9911061324
//9911021733 User data may contain
//comma, continuous blank, newline
//tab etc. The code below change
//data string all be separate by
//one blank. Ready for .split(" ") 
//xycoord=inStr1.split(" "); //9910302212
//change number string to array.
//
//First change newline to a " ".
var i,j,k;
var wk1="";
for(i=0;i<arg1.length;i++)
  {
//Read new line byte "\n"
if(arg1.charAt(i)=="\n")wk1+=" ";//change to " "
//Read to return to beginning byte "\r"
else if(arg1.charAt(i)=="\r")wk1+="";//drop it.
else wk1+=arg1.charAt(i); //9910302127 copy other bytes.
  }
//Change all comma to blank
wk1=wk1.replace(/,/g," "); //9910302139
//Change all tab to blank
wk1=wk1.replace(/\t/g," "); //9910302140
//Change all two blank to one blank
while(wk1.indexOf("  ")>=0)
wk1=wk1.replace("  "," "); //9910302159

//Drop string leading blank
j=0; //9911071210
while(wk1.charAt(j)==" ")j++;
//if string has leading blank and
//not drop it , inStr1.split(" ");
//will shift data and x data become 
//y data, y data become x data.
//horizontal sine wave stand up !!

//drop string ending blanks.
//k=wk1.lastIndexOf(" ")
//if(k==-1)k=wk1.length;
//Above is ill. wipe out last data.
k=wk1.length-1;
while(wk1.charAt(k)==" ")k--;
k++; //9911030003

//wk1=wk1.substring(0,k); //9910302210
wk1=wk1.substring(j,k); //9911071212
return wk1;
} //function cleanData(arg1) //9911061326













//vector v1 cross product vector v2
function v1Xv2(v1,v2) //9712311019
{
  var v3=new Array(3);
v3[0]=v1[1]*v2[2]-v1[2]*v2[1]
v3[1]=v1[2]*v2[0]-v1[0]*v2[2]
v3[2]=v1[0]*v2[1]-v1[1]*v2[0]
  return v3; // 9712311023
}





// find vector length, good for 2D,3D,4D...
function length1(inVec) //9712311031
{
  var ii,jj=0;
  var siz0=inVec.length; //9712311032
  for(ii=0;ii<siz0;ii++)
  {
  if(isNaN(inVec[ii]))return -1;
  jj+=inVec[ii]*inVec[ii];
  }
  return Math.sqrt(jj);
} //9712311035





function vec1(q)
{ //a006240918 normalize vector
var i,j,k;
if(typeof(q)=='string')
  { 
if(q.indexOf(',')>0)q=q.split(',');
else
    {
while(q.indexOf('  ')>=0)
q.replace('  ',' ');
var i=0;var j=q.length-1;
if(q.charAt(i)==' ')i++;
if(q.charAt(j)==' ')j--;
q=q.substring(i,j+1);
q=q.split(' ');
    }
  }
var L=Math.sqrt(q[0]*q[0]
+q[1]*q[1]+q[2]*q[2]);
if(Math.abs(L)>1.e-8)
q=[q[0]/L,q[1]/L,q[2]/L]
return q;
} //function vec1(q)  //a006240926












/**
function dotPrecision() 
read three data type
(1) number, example  1.2345678
(2) string, example "1.2345678"
(3) number array, example
    [1.2345678909876543,9.87654321234567]

if input is string array
 ["3.654678","4.535366"] 
dotPrecision() return as is.

 first argument arg1 is a set of data
second argument arg2 is a positive ingeter
if arg2=3 , then input data 1.23456789
cut short to 1.235 . (has rounding)
input data "1.2345678909876543"
cut short to "1.234" .(no rounding)

2011-02-26-11-17 start
This version remove one error
function dotPrecision()
if input is
dotPrecision([+1.1548,-1.1548,3.21])
old version correct rounding +1.155 
old version  error  rounding -1.154
correction made at a002261037
2011-02-26-11-18 stop

correction made at a003061903

/**/
function dotPrecision(arg1,arg2)
{ //9911171912

// arg2 is a positive integer
//represent how many digit to keep
//after dot
var precisionTo=parseInt(arg2); 
if(isNaN(precisionTo)
 ||precisionTo<0 ||precisionTo>15)
//default to keep 3 digits
  precisionTo=3; //9911101321

var usedigit=Math.pow(10,precisionTo);

if(typeof(arg1)=='number')
  { //9911201132
if(arg1>=0)
return parseInt(arg1*usedigit+0.5)/usedigit;
else //a003061903
return parseInt(arg1*usedigit-0.5)/usedigit;
  }
//2011-03-06-19-06 document stop
//2011-03-06-20-38 document start

var inType=0
var inStr6;
var ouStr6;
var inArr6;
var ouArr6;

if(typeof(arg1)=='string')
  {
inStr6=arg1;
ouStr6='';
inType=1; //9911171918
  }
else
if(typeof(arg1)=='object'
 &&typeof(arg1[0])=='number'
  )
  {
inArr6=arg1;
ouArr6=[];
inType=2; //9911171922
  }

var i,j,k;

if(inType==1)  //inpur is string
  { //for example inStr6="3.654678 4.535366"
for(i=0;i<inStr6.length;i++)
    {
//if no decimal point, skip integer
if(inStr6.charAt(i)!='.')
      {
ouStr6+=inStr6.charAt(i);
continue;
      }
//has decimal point
//copy decimal point
//and begin to work
//"3.654678 4.535366"
//Now work point is 6 in 3.6
ouStr6+=inStr6.charAt(i);
i++; //9911101344
//this is j loop, but for(j) body no j
//because j loop control precisionTo=3 times
for(j=0;j<precisionTo;j++)
      {
//if string is "3.6  4.535366"
//read ' ', not digit, stop
if(!isDigit(inStr6.charAt(i)))
break; //9911102154
//read digit, continue read
ouStr6+=inStr6.charAt(i);
//counter i increase one
//(not j increase one)
i++; //9911101347
      } //for(j=0;j<precisionTo;j++)

//input string "3.654678 4.535366"
//record up to 3.654 and stop, because
// j loop control precisionTo=3 times
//next while() loop discard "678" 
while(isDigit(inStr6.charAt(i)))
i++; //9911101345

//input string "3.654678 4.535366"
//where "8 4" has blank, not digit
//when while() stop, i point to ' '
//in "8 4" . Blank is copied
//save ' ' to output ouStr6
ouStr6+=inStr6.charAt(i); //9911101331
    } //for(i=0;i<inStr6.length;i++)

//input "3.654678 4.535366"
//return "3.654 4.535"
return ouStr6; //9911171915
  } //if(inType==1)

//Above input string "3.654678 4.535366"
//Below number array [3.654678,4.535366]
if(inType==2)
  { //9911171926
//variable ptfive take care 
//four drop five enter (rounding)
var ptfive=0.5; //a002261036
for(i=0;i<inArr6.length;i++)
    {
ptfive=0.5;
if(inArr6[i]<0)ptfive=-0.5; //a002261037
ouArr6[i]= //9911171928
parseInt(inArr6[i]*usedigit+ptfive)/usedigit;
    } //for(i=0;i<inStr6.length;i++)
return ouArr6; //9911171929
  } //if(inType==2)

return arg1; //9911171931

} //function dotPrecision(arg1) 












/**
9912130920 in xygraph3.htm change
function bye09() to bye09v02()

function bye09(in09) has error
bye09(-0.99999999999) return +1 
but it should return -1
bye09(-.99999999999)  return  0 
but it should return -1

function bye09v02(in09) output correctly
9912122305

2009-06-12-15-31 start
goodbye09  bye09v02()
Javascript use double precision
for real number calculation
Answer may have long string
'000000' or '999999', example
2.00000000000001 
-25.999999999999996 
on 2009-06-12-11-58 Liu,Hsinhan
write a function
function bye09(in09) //9806121158
call bye09v02(in09) as following
coef2=bye09v02(coef2);
where coef2 is a number, after call
change from 2.00000000000001 to 2
change from -25.999999999999996 to -26
This is a handy tool, may be 
you like this function.
if number is 1.23111111111 
no 0000000 no 9999999 ,
return as is.

function bye09v02(in09) modify number
for easier reading. During calculation
please do not call bye09v02() that is
just slow down speed.
If know answer do not conatin 0000000 
and do not contain 9999999 ,
no need call bye09v02().

If expect answer is integer or short
float number. for example 1.2 in 
this case you can call bye09v02().
2009-06-12-15-40 stop
//2010-07-27-09-47 stop


//2010-07-27-10-07 start
2009-06-17-19-05
if you have number array, do not send
whole array to bye09v02() Because
bye09v02() one time work on one number

This file has manager file example ,
Please find '9806171858'
2009-06-17-19-09 stop
//2010-07-27-10-11 stop
/**/
function bye09v02(in09) 
//9806121158 v01,
//9912122230 v02
{ // in09 is input number
// for example 2.0000000001
// if if it is not a number, return
if(isNaN(in09)) return in09;

// where is '.' ?
var dotLoc=(in09+'').indexOf('.',0);
// no '.', integer, return
if(dotLoc==-1) return in09;

// whether number has '000000' ?
var zeroLoc=(in09+'').indexOf('000000',dotLoc);
// find '000000' cut '000000' 
if(zeroLoc>0) //9806121216
return parseFloat((in09+'').substring(0,zeroLoc)); //9806121209

// whether number has '999999' ?
var nineLoc=(in09+'').indexOf('999999',dotLoc);
// find '999999' , begin cut '999999'
// need find rounding
if(nineLoc>0) //9806121223 
  { //cut at left of '999999' 

// change inpot to number
in09=parseFloat(in09);

var nine1;   //9912122231
var tiny6=Math.pow(10,-(nineLoc-dotLoc+5)); //9912122232
if(in09>0)nine1=in09+tiny6;
else      nine1=in09-tiny6;  //9912122240

//next line has '000000' 
return bye09v02(nine1); //9912122247

  }
// no '000000' no '999999' return as is
return parseFloat(in09); //9806121224
} //function bye09v02(in09) 
//2010-07-27-10-25 stop, bye09()
//2010-12-12-22-52 stop, bye09v02()









//2010-07-26-19-18
//bye09a() = bye09v02() for Array
//bye09a() do not do bye09v02() work
//bye09a() read array number and 
//call bye09v02() to work. You must
//have bye09v02() in same file
//
function bye09a(arg1) //9907261918
{
if(typeof arg1=='number')
return bye09v02(arg1);  //9907261920

if(typeof arg1!='object')
return arg1;  //9907261921

if(typeof arg1[0]=='object'
 &&typeof arg1[0][0]=='object'
  )
return arg1;  //9907261922

var i,j,k;
var oup0='';

if(typeof arg1[0]=='object'
 &&typeof arg1[0][0]=='number'
  )
  {
for(i=0;i<arg1.length;i++)
for(j=0;j<arg1[i].length;j++)
arg1[i][j]=bye09v02(arg1[i][j]);
  }
else
if(typeof arg1[0]=='number')
  {
for(i=0;i<arg1.length;i++)
arg1[i]=bye09v02(arg1[i]);
  }
return arg1;
} //function bye09a(arg1) //9907261926









//a004081920 start
//s2aEZ(arg1,'text') second parameter
//no need to be 'text'
//when s2aEZ() see two input parameters
//then carry out text equation
//that is, not run eval(string)
//Ask output to be text, 
//why change sin(1.2) to 0.93203 ?
//a004081925 stop

//a002271137
//function s2aEZ(inString,notParse)
//do simple string to array change
//for example, input number string
//"1.2,3.4,5.6\n7.8,9.0,-2.1"
//output next number array
//[[1.2,3.4,5.6],[7.8,9.0,-2.1]]
//a002271139
//
//a003162025 add for(j) loop
//for(j=0;j<ouArr[i].length;j++)
//    { //a003162025
//ouArr[i][j]=parseFloat(ouArr[i][j]);
//    }
//return float number, 
//not return string number. a003162033
//
//a003171949 add notParse capability
//when call s2aEZ() if use two parameters
//output string, not output number array
//then process symbolic equation cosψcosθ 
//is possible.
//
function s2aEZ(inString,notParse)
{ //a002271131 
if(typeof(inString)!='string')
return inString;

/**
a003231050
function s2aEZ(inString,notParse) 
change string type matrix to 
string array type matrix.

input: string type matrix
"1,4,7,9,6,3\n
 2,5,8,8,5,2\n
 3,6,9,7,4,1"
in which [\n] is newline byte.

two arguments call 
s2aEZ(inString,'notParse') 
output: string array type matrix
[['1','4','7','9','6','3'],
 ['2','5','8','8','5','2'],
 ['3','6','9','7','4','1']]

one arguments call 
s2aEZ(inString) 
output number array type matrix
[[1,4,7,9,6,3],
 [2,5,8,8,5,2],
 [3,6,9,7,4,1]]

function s2aEZ() consider comma as
a data separater in a row.
function s2aEZ() consider newline as
a row separater in a matrix.

comma tell function s2aEZ() there
are more data follow. Therefore,
last data in one row can not have
a comma after it.

newline tell function s2aEZ() there
are more row follow. Therefore,
last row in one matrix can not have
a row after it.

User may put many newline at the 
end of matrix, for example
[[
1,4,7,9,6,3\n
2,5,8,8,5,2\n
3,6,9,7,4,1\N
\N
\N
]]
Those uppercase [\N] all mislead 
function s2aEZ() and cause error.
function s2aEZ() first job is to 
delete the matrix end newlines.
a003231102
/**/

var i,j,k,m,n; //a003162024

//a004221009 start
//newline byte is control byte. All
//control byte ascii number are less 
//than 32. next code
//Let i point to string end
i=inString.length-1;
//if this byte is control byte
while(inString.charCodeAt(i)<32) //a003231113 while()
//index i reduce by one
i--;
//while() stop at a place over-reduce
//by one. recover this one.
i++; 
//copy original string, start from 0
//to index i， net result is discard
//line end newline bytes (tab too)
inString=inString.substring(0,i);
//a004221018 stop

var inArr=[]; 

if(inString.indexOf('\n')>0) //a003162320 if()
inArr=inString.split('\n');

//if input do not have new line
//store input to ouArr[0]
else inArr[0]=inString; //a003162359

var ouArr=[];
for(i=0;i<inArr.length;i++)
  { //a002271133

if(inArr[i].indexOf(',')>0) //a003162340 if()
    {
//split one row data at ','
ouArr[i]=inArr[i].split(',');
    }

//split one row data at tab
else //a003170022
if(inArr[i].indexOf('\t')>0)
ouArr[i]=inArr[i].split('\t');

//split one row data at ';'
else //a003220007
if(inArr[i].indexOf(';')>0)
ouArr[i]=inArr[i].split(';');

//split one row data at ' '
else
    {

k=inArr[i]; //a003162326 while();
while(k.indexOf('  ')>=0)k=k.replace('  ',' '); //a003250952 >=0 ('  ',' ')


//drop string leading blank
//drop string ending blank
//a003250928
m=0;
if(k.charAt(0)==' ')m=1; //a003250930
n=k.length-1;
if(k.charAt(n)==' ')n--; //a003250931
k=k.substring(m,n+1);    //a003250932
//
//code at a003250952 already
//change all '  ' to ' ', then
//leading blank, ending blank
//have at most one ' '. a004221307


/**
a003231125 
if input one column
"1
2
3
4
5"
after remove leading/ending ' '
one whole row is (for example)
"3"
no ' ', following code 
if(k.indexOf(' ')<0)
become true. but
"1
2
3
4
5"
is a column vector, it 
should be stored as
m[0][0]='1'
m[1][0]='2'
m[2][0]='3'
m[3][0]='4'
m[4][0]='5'
following
ouArr[i]=[]; //a003231123
declare ouArr[i] is array
of one more level,
after declare, allow to use 
ouArr[i][0]
this time, store k to ouArr[i][0]
ouArr[i][0]=parseFloat(k);
a003231131
/**/
if(k.indexOf(' ')<0) //a003221618
      {
ouArr[i]=[]; //a003231123
ouArr[i][0]=k;
      }
else
//othwise one row has ' '
//split data at ' '
ouArr[i]=k.split(' ');
    } //if(inArr[i].indexOf(',')>0)
      //else '\t' else ';' else ' '

//if input symbolic matrix cosψcosθ 
//skip parseFloat()
if(arguments.length>1)
    {
//a003192028 if input numerical matrix
//use parseFloat() auto drop '\n'
//if input symbolic matrix, 
//not use parseFloat(), not drop '\n'
//cause trouble.
//when if(arguments.length>1) be true
//input symbolic matrix, next code
//ouArr[i][j].substring(0,ouArr[i][j].length-1)
//drop '\n' 

j=ouArr[i].length-1; //a003192031
if(ouArr[i][j].charCodeAt(ouArr[i][j].length-1)<32) //a003192213 if()
ouArr[i][j]=ouArr[i][j].substring(0,ouArr[i][j].length-1) //a003192033

continue; //a003171951
    }

if(typeof(ouArr[i])=='object') //a003221628 if([i])
//change string '2' to number 2
for(j=0;j<ouArr[i].length;j++)
    { //a003162025
ouArr[i][j]=parseFloat(ouArr[i][j]);
    }
  } //for(i=0;i<inArr.length;i++)

//if input one row, return ouArr[0]
if(ouArr.length==1)
  {
return ouArr[0]; //a003170951
  }

return ouArr; //a002271136
} //function s2aEZ(inString) 











//9911182053
var W3CcolorTable=[
['#00FFFF','aqua','水藍色'],       //0
['#000000','black','黑色'],        //1
['#0000FF','blue','藍色'],         //2
['#FF00FF','fuchsia','櫻紅色'],    //3
['#808080','gray','灰色'],         //4
['#008000','green','深綠色半綠色'],  //5
['#00FF00','lime','綠色'],         //6
['#800000','maroon','褐紅色半紅色'], //7
['#000080','navy','海藍色半藍色'],   //8
['#808000','olive','橄欖色'],       //9
['#800080','purple','紫色'],       //10
['#FF0000','red','紅色'],          //11
['#C0C0C0','silver','銀色'],       //12
['#008080','teal','深水色'],        //13
['#FFFFFF','white','白色'],        //14
['#FFFF00','yellow','黃色']        //15
]
;
//9911182059



//a006171200 自 xygraph1.htm 
//取用 function cn2color(arg1) 
//存於 sphtrian.htm
//
//9911031104
//9911231101
function cn2color(arg1)
{
if(arg1=='')return '#000000';

var cnColor; //9911231102
var cnCindx;

if(typeof(colorCN)=='object')
  {
cnColor=colorCN;
cnCindx=1;
  }
else
if(typeof(W3CcolorTable)=='object')
  {
cnColor=W3CcolorTable;
cnCindx=2;
  } //9911231104

var i,j,k;

for(i=0;i<cnColor.length;i++)
  {
j=cnColor[i][cnCindx].indexOf(arg1); //9911231106

if(j>=0&&j!=1)
return cnColor[i][0];
  }

return arg1;
} //function cn2color(arg1)
//9911031108
//9911231107









/**
a002281239
2011-02-28-12-39
http://freeman2.com/xygraph3.htm and 
http://freeman2.com/xygraph4.htm have
function runTestCode(arg1) //9911231003
require arg1 be 0,1,2,3
it use next code
[[
if(arg1==0)
inStr1=QSboxb01.value; //9912221700
else
if(arg1==1||arg1==2)
inStr1=QSboxb03.value; //9911231331
else
if(arg1==3)
inStr1=QSboxb04.value; //9912301133
]]

http://freeman2.com/tutc0054.htm and 
http://freeman2.com/tute0054.htm have
function runTestCodeQF(arg1)
LiuHH change function name 
from runTestCode(arg1)
to   runTestCodeQF(arg1)

call function runTestCodeQF(arg1) as
runTestCodeQF(QFboxd13.value)
here arg1 is not 0,1,2,3
arg1 is QFboxd13.value
2011-02-28-12-50

2011-05-27-10-41 change QF to QA

//9911231003 QS xygraph4.htm 9912120858
//a002271621 QF tute0054.htm
//a005271041 QA axs2mat2.htm a005271046 run OK
//a007041911 QA sphtrian.htm 
//
/**/
function runTestCodeQA(arg1)
{
//9912292223 move next two if() to top
//9912292225 ok right
if(typeof(xyatL)=='function') //9912292210
xyatL(QAxatL,QAyatL); //9912292211

var MyGraph = new XYGraph();
var MyLine1 = new XYLine();
MyLine1.VMLpointshapetype="none";
if(typeof(QAdivGraph10)=="object")
QAdivGraph10.innerHTML = ""; 

	MyGraph.ymax=QAboxYmax0.value;
 MyGraph.ymin=QAboxYmin0.value;
	MyGraph.xmax=QAboxXmax0.value;
 MyGraph.xmin=QAboxXmin0.value;
	MyGraph.gheight=QAboxHeight0.value;
 MyGraph.gwidth=QAboxWidth0.value;

showXYcoord1orNot0=1;

if(typeof(QAchkbxID51)=='object')
  {
if(!QAchkbxID51.checked)
showXYcoord1orNot0=0; //9912172128

//a007041916 move to here
if(typeof(xyaxis01)=='function') 
xyaxis01(QAchkbxID51); //9912292212
  }

MyLine1.x=[0,0];
MyLine1.y=[0,0];
MyGraph.Plot(MyLine1); 
if(arg1==0) //9912120909
  {
QAdivGraph10.innerHTML = MyGraph; 
return; //9912120910
  }

//Above code is common to all parts.
//
//

var inStr1=arg1; //9912121145

var inStr2;
var bgn1,end1;
bgn1=inStr1.indexOf('//[[[')
end1=inStr1.indexOf('//]]]')
if(bgn1>0&&end1>0)
inStr2=inStr1.substring(bgn1,end1)
else
inStr2=inStr1;
eval(inStr2); //9911231335

//
//
//All line code, label code must be above here.
//must be before next "... = MyGraph" line.
QAdivGraph10.innerHTML = MyGraph; 

} //function runTestCodeQA(arg1) 9912120915






//-->
</script><!--[=][][]-->


<br><hr><hr><br>


<pre><font size=+2><a name="docB_001">&lt;a name="docB_001"&gt;</a>
<script language="javascript">link1()</script>
2011-07-06-14-11 start
This file sphtrian.htm is a utility
to draw spherical triangle. It was 
a big file, include this file QADraw
and tute0055.htm QBDraw, QCDraw and
QDDraw. Because file size exceed 
500MBytes, decide to split to two
files tute0055.htm and sphtrian.htm 
sphtrian has link with tute0055.

<a name="docB_002">&lt;a name="docB_002"&gt;</a>
Following is QADraw control panel
[[<a href=#QADraw>link</a><font color=red size=+3>
Spherical triangle graph page</font>
Graph area size, W:[ ] H:[ ]      
x min:[ ], x max:[ ]; y min:[ ], y max:[ ]; 
<font color=red>x,y value below should be within range above</font> <input type="button" value="-1to1" /> 
]]
Above control drawing board 
dimension and x/y min/max range.
Click <input type="button" value="-1to1" /> or <input type="button" value="-3to3" /> program update 
x/y min/max range. If need other 
range, change by hand.

<a name="docB_003">&lt;a name="docB_003"&gt;</a>
</font></pre>
[[<a href=#lmnStar>link</a>
<br>
Sph.triangle abc, given a,b and angle a & b; c is unknown. 
<br>
vector a≡l,b≡m,p=l×m have auxiliary vector l*,m*,p*. 
<br>
l,m,p all given, see <a href=tute0055.htm#a055_132>tute0055.htm</a> eqn.rod136 to 139. 
<br>
Draw l*,m*,p* or not ? □not draw l*,m*,p*
<br>
<font color=red>
l*,m*,p* length may be>1. Please increase x/y range. 
</font> <input type="button" value="-3to3" /> 
<br>
If draw l*,m*,p* click "Draw triangle AB side & angle" 
<br>
or click "reverse". If click "Draw triangle ABC sides" 
<br>
program ignore l*,m*,p* and not draw them. 
<br>
l*,m*,p*color=[ ], width=[ ], solidot=[ ] 
<br>
]]
<pre><font size=+2><a name="docB_004">&lt;a name="docB_004"&gt;</a>
Vectors a≡l,b≡m,p=l×m are given. 
Their auxiliary vectors are
l*,m*,p* Code to draw auxiliary 
was added after upload tute0055
mainly Liu,Hsinhan hope to know
the relative location between 
given l,m,p and unknown l*,m*,p*
Default □not draw l*,m*,p* If 
click ▣yes draw l*,m*,p* and if
triangle area is too small, in 
this case aux. vectors length
are greater than  x/y [-1,+1] 
Please click <input type="button" value="-3to3" />, see whether 
help, if not, hand adjust
x/y min/max range.

<a name="docB_005">&lt;a name="docB_005"&gt;</a>
If given three vertex vectors 
of a triangle or if given two
vertex vectors and two vertex
angles, both define a triangle.
This file has two input methods
First input three vertex vectors 
click button is <input type="button" value="Draw triangle
ABC sides" /> 
Second input two vertex vectors 
and two vertex angles. Click 
button is <input type="button" value="Draw triangle
AB side & angle" /> and <input type="button" value="reverse" />
<a name="docB_006">&lt;a name="docB_006"&gt;</a>
<script language="javascript">link1()</script>
First input method has no thing
to do with proving Euler's 
rotation theorem.
Second input method relate to 
proving Euler's rotation theorem.
Aux. vectors l*,m*,p* is used in
proving Euler's theorem. If click
▣yes draw l*,m*,p* and if click 
[Draw triangle ABC sides] user 
will not see l*,m*,p* graph. 
Because program not care l*,m*,p*
In this case, click
[Draw triangle AB side & angle] 
or [reverse] to get l*,m*,p* graph.

<a name="docB_007">&lt;a name="docB_007"&gt;</a>
l*,m*,p* color=[ ], width=[ ], solidot=[ ] 
determine aux. l*,m*,p* line 
color/width/solidot property.

OA draw ▲,OB draw ◆,OC draw ●.
l* draw ▲,m* draw ◆,p* draw ×.
p* not relate to OC, therefore 
p* not draw ●.

Liu,Hsinhan think the need of
drawing aux. l*,m*,p* is minimum,
and put its control box far away
from drawing board.
2011-07-06-15-03 write note here
2011-07-09-22-17 translate here

<a name="docB_008">&lt;a name="docB_008"&gt;</a>
[[<a href=#QAground>link</a>
ground (fixed) 3D triad 
color=[ ], width=[ ], solidot=[ ] 
recover default color/width/solidot click <input type="button" value="CWS" /> CWS 
]]
Let user decide ground triad 
color/width/solidot.

<a name="docB_009">&lt;a name="docB_009"&gt;</a>
[[<a href=#QAground>link</a>
b,c color=[ ], width=[ ], solidot=[ ] 
c,a color=[ ], width=[ ], solidot=[ ] 
a,b color=[ ], width=[ ], solidot=[ ] 
a,b,c color=[ ], width=[ ], solidot=[ ] 
triangle color=[ ], width=[ ], solidot=[ ] 
]]
[b,c] indicate great circle pass 
vector b and c. Above boxes let 
user assign three great circle 
color/width/solidot and three 
radius OA,OB,OC and triangle ABC
color/width/solidot.

<a name="docB_010">&lt;a name="docB_010"&gt;</a>
[[<a href=#QAground>link</a>
tangent color=[ ], width=[ ], solidot=[ ] 
Draw two tangent at vertex A▲□; B◆□; C●□ 
]]
Let user assign vertex two 
tangents color/width/solidot.
To prove spherical triangle 
cosine law need vertex two 
tangents. Please see <a href=tute0055.htm#QCDraw>tute0055</a>
QCDraw <input type="button" value="fig.1" /> 
If click A▲▣; B◆□; C●□ 
Only vertex A draw tangents.
If click A▲▣; B◆▣; C●▣ 
Three vertex A,B,C all draw 
tangents. Program did not stop
user from choosing multiple 
vertex. Default draw none.

<a name="docB_011">&lt;a name="docB_011"&gt;</a>
<script language="javascript">link1()</script>
[[<a href=#QAground>link</a>
Axes cross at x=[ ], y=[ ], 2 0 ;□Del x,y axis 
]]
To draw spherical triangle, not
need x,y coordinates.
Default: □Del x,y axis 
If click ▣Use x,y axis then 
"Axes cross at x=[ ], y=[ ]"
is effective. Most case set at
x=0, y=0. But if graph range 
is x/y = 10/20, in this case 
set at x=15, y=15. If not, the
range x/y = 10/20 become 
x/y = 0/20. Because XYGraph 
insist to display axes cross 
point on display.
2011-07-06-15-50 here

<a name="docB_012">&lt;a name="docB_012"&gt;</a>
[[<a href=#axisAnglBox>link</a></font></pre>
Next five boxes convert φ,θ,ρ to x,y,z use degree. 
<br>
Three numbers are latitude,longitude,radius. 
<br>
Equator is latitude=0. If click south/north be ▣ polar 
<br>
latitude place read as polar angle. North pole=0. 
<br>
If fill in just two numbers, program set radius=1. 
<br>
"I" is eye ; "U" is up ; "A" is vertex A 
<br>

<input type="button" value="d" /> 
<input id=QAZboxEyeRho value="0,12.5,1"
 size=20
 />

<input type="button" value="I" /> ;

<input type="button" value="d" /> 
<input id=QAZbox_UpRho value="0,0,1.2"
 size=20
 />

<input type="button" value="U" /> ;

<br>

<input 
 type="button" value="d" /> 
<input id=QAZboxARho value="16,75,1.0"
 size=20
 />

<input type="button" value="A" /> 
 , 

<input type="button" value="d" /> 
<input id=QAZboxBRho value="48,-18,1.0"
 size=20
 />

<input type="button" value="B" /> 

<br>

<input type="button" value="d" /> 
<input id=QAZboxCRho value="36,8,1.0"
 size=20
 />

<input type="button" value="C" /> 


 south/north be <input
 type="checkbox"
 id="QAZchkbxID53"
 onclick="javascript:if(QAZchkbxID53.checked)
QAZspanAlert53.innerHTML='polar '
else
QAZspanAlert53.innerHTML='latitude '
"
 /> 
<span id=QAZspanAlert53>
latitude 
</span>
<br>
Program do not read above five boxes, read only box below 
<pre><font size=+2>]]
Program require input to be (x,y,z).
If user has latitude/longitude in
hand, fill data to above boxes 
(choose a right one) Click right 
side button to convert.

<a name="docB_013">&lt;a name="docB_013"&gt;</a>
[[<a href=#QAeyeUpbox>link</a>
View triangle at center: build data after 1st draw
eye xyz <input id=QAZboxCRho value="36,8,1.0"
 size=20
 /> <input type="button" value="dI" /> <input type="button" value="center" /> 
Up xyz <input id=QAZboxCRho value="36,8,1.0"
 size=20
 /> <input type="button" value="dU" /> <input type="button" value="use8" />
]]
Draw 3D object on 2D monitor, 
must assign eye location and 
screen up direction. Fill eye 
and up data to above boxes.
If eye is at side of triangle 
it is hard to compare three 
angles relative magnitude. 
When draw first time, program
calculate triangle center x,y,z
<a name="docB_014">&lt;a name="docB_014"&gt;</a>
View triangle at center: build data after 1st draw
and change above to next
View triangle at center: 0.506,0.61,0.539
Now click <input type="button" value="center" /> program read [center
 x,y,z] and fill to eye xyz box.
Draw second time, eye is on 
center view. See relative angle
better. Two input methods:
1: given vertex A,B,C. It is 
   possible to calculate center
   x,y,z without first run.
2: given vertex A,B and angle 
   A,B. It is not possible to 
   calculate center x,y,z without 
   first run.
This program is designed after 
first draw get center x,y,z.

<a name="docB_015">&lt;a name="docB_015"&gt;</a>
[[<a href=#QAeyeUpbox>link</a></font></pre>
Click R, Record data to box17, auto ▣light; 
<br>
Click A, Apply box17 data; 
<br>
Click D, Delete box17 data, auto □light; 
<br>
Save box17 data to your file for future use. 
<br>

<input type="button" value="dABC" /> 
<input type="button" value="uABC" /> 
 ; 
<input type="button" value="dAB" /> 
<input type="button" value="uAB" /> 
; <a href="#QAbox17">record</a> 
<INPUT type=button value="R">
<INPUT type=button value="A">
<INPUT type=button value="D">

<input
 type="checkbox"
 id="QAZchkbxID71"
/>light  

<pre><font size=+2>]]
<a name="docB_016">&lt;a name="docB_016"&gt;</a>
<script language="javascript">link1()</script>
After created QBDraw,QCDraw,
Liu,Hsinhan hope to find their 
original 3D data, but not keep
and disappear. Then added new 
buttons <INPUT type=button value="R"><INPUT type=button value="A"><INPUT type=button value="D"> for future work 
archive useful 3D data. If user
satisfy a graph output, click <INPUT type=button value="R"> 
to Record all 3D data to Box17.
User goto box17 to copy and save
data for future re-use.
<a name="docB_017">&lt;a name="docB_017"&gt;</a>
When re-draw, copy 3D data and 
paste to Box17, then click <INPUT type=button value="A"> to 
Apply data to right input boxes,
click [Draw triangle ABC sides] 
or Draw triangle AB side & angle]
or [reverse] to get old graph.
If click <INPUT type=button value="R">, light become [▣light]
If click <INPUT type=button value="D">, light become [□light]
indicate whether Box17 has data.
If click [□light] program goto 
Box17 and see if there is data 
or not, then update light.

<a name="docB_018">&lt;a name="docB_018"&gt;</a>
[[<a href=#ABCinBox>link</a></font></pre>
A,B,C is spherical triangle three vertex vectors. 
<br>
Given A,B (but not C) and angle A,B fill in next four boxes 
<br>
angle must be degree, program do not cut angle to half. 
<br>

angle:<input value="24" size=20 /> ; angle:<input value="36" size=20 /> 
<br>
Given A,B,C (but not angles), fill in next three boxes 
<br>
A:<input value="1.1,2.2,3.3" size=20 /> ; B:<input value="2.4,3.5,2.1" size=20 />  
<br>
C:<input value="3.6,2.7,1.8" size=20 /> A,B,C length will be normalized 
<br>
QADraw <input type="button" value="Draw triangle
ABC sides" /> ; <input type="button" value="Draw triangle
AB side & angle" /> <input type="button" value="reverse" /> ; <input type="button" value="wipe" />
<br>
[ drawing board is here ]
<br>
<pre><font size=+2>]]
<a name="docB_019">&lt;a name="docB_019"&gt;</a>
2011-07-06-16-32 here
Above is QADraw main input boxes.
If know triangle A,B,C three 
vectors, fill them to A,B,C 
three boxes.
If know triangle A,B two vectors
and A,B two angles, fill them to 
A,B vector and angle four boxes.
Angle box must use degree, not 
radian, and must not rotation 
angle full angles. User divide 
rotation angle by two.
2011-07-06-16-37 stop<!--中文變方格，關機-->
<a name="docB_020">&lt;a name="docB_020"&gt;</a>
2011-07-06-18-06 start
Program will not divide rotation 
angle by two. If program does, 
then all non-rotation user must
multiply vertex angle by two 
before fill in values.
Rotation user fill in half angle
in degree.
If click <input type="button" value="Draw triangle 
AB side & angle" /> program run
 qC=quatquat(qA,qB);
If click <input type="button" value="reverse" /> program run
 qC=quatquat(qB,qA);
"reverse" say 
quaternion multiplication
 qA*qB reverse to qB*qA.
<a name="docB_021">&lt;a name="docB_021"&gt;</a>
<script language="javascript">link1()</script>
Related code is next
function drawSphereTriangle()
{
.....
if(arguments.length==4)
qC=quatquat(qA,qB); //a006161606 
else
qC=quatquat(qB,qA); //a006171855
.....
}

<a name="docB_022">&lt;a name="docB_022"&gt;</a>
[[<a href=#QAbox11>link</a>
<input type="button" value="%% draw Box11 code" /> 
Box11, output XYGraph code 
<TEXTAREA id=QAZboxc11 rows=3 cols=50 ></TEXTAREA>
]]
Box11, output XYGraph code. On
the other hand XYGraph code in 
Box11 can be tested by click
<input type="button" value="%% draw Box11 code" /> .
<a name="docB_023">&lt;a name="docB_023"&gt;</a>
If hope to display original 
triangle and dual triangle at
same time. Run original once,
keep its XYGraph code. Then 
run dual once, box11 has dual
code. Now paste original code
to end of (or begin of) box11.
With two codes in one box, 
click <input type="button" value="%% draw Box11 code" />
output both triangles.

<a name="docB_024">&lt;a name="docB_024"&gt;</a>
[[<a href=#QAbox12>link</a>
Box12, output 2D x,y data 
<TEXTAREA id=QAZboxc12 rows=3 cols=50 ></TEXTAREA>

Box13, output 3D x,y,z data 
<TEXTAREA id=QAZboxc13 rows=3 cols=50 ></TEXTAREA>

Box14, vector,angle,area,normal
<TEXTAREA id=QAZboxc14 rows=3 cols=50 ></TEXTAREA>
 
Box15, rotation matrix 
<TEXTAREA id=QAZboxc15 rows=3 cols=50 ></TEXTAREA>
]]
<a name="docB_025">&lt;a name="docB_025"&gt;</a>
Box12 to Box15 output function
are marked above each box. In 
which [normal] may be useful.
Draw triangle vertex tangent 
lines, with arbitrary eye xyz 
it is not easy to see the right
angle between tangent & radius.
Run [Draw triangle ABC sides]
Box14 output normal vectors.
Choose a right normal and paste
to eye xyz box. Run second time
to see the right angle better.
But view triangle from side, 
expect to see triangle thin as
a paper side view.
2011-07-06-18-56 stop

<a name="docB_026">&lt;a name="docB_026"&gt;</a>
<script language="javascript">link1()</script>
2011-07-06-19-56 start
[[<a href=#QAbox15>link</a></font></pre>

<input type="button" value="EyeUp Mat" /> 

<input type="button" value="31" /> 
<input type="button" value="32" /> 
<input type="button" value="33" /> 
; 
<input type="button" value="transpose" /> 

<input type="button" value="21" /> 
<input type="button" value="22" /> 
; 
<input type="button" value="34" /> 
to ABC <a href="#ABCinBox">box</a>

<br>

<input type="button" value="read" /> 

<input type="button" value="dual triangle" /> 

<INPUT name="QAZRadio1" type="radio"
>1 
<INPUT name="QAZRadio1" type="radio"
>2 
<INPUT name="QAZRadio1" type="radio"
>3 
<INPUT name="QAZRadio1" type="radio"
>4 

<input type="button" value="fill" />

<INPUT name="QAZRadio1" type="radio"
>5 
<INPUT name="QAZRadio1" type="radio"
>6 
<INPUT name="QAZRadio1" type="radio"
>7 
<INPUT name="QAZRadio1" type="radio"
 checked
>8 
<pre><font size=+2>]]
Above buttons main function is 
debug program functions.
<a name="docB_027">&lt;a name="docB_027"&gt;</a>
<input type="button" value="EyeUp Mat" /> test function eyeUpMat() 
This file <a href="#docA_001">top</a> has document. 
Please click <input type="button" value="31" /><input type="button" value="32" /><input type="button" value="33" /> Box15 build 
examples. Then click <input type="button" value="EyeUp Mat" /> 
output to Box16.

<input type="button" value="transpose" /> test function transpoMat()
Please click <input type="button" value="21" /><input type="button" value="22" /> Box15 has example 
input, then click <input type="button" value="transpose" /> Output to 
Box16.

<a name="docB_028">&lt;a name="docB_028"&gt;</a>
Based on original vertex, <input type="button" value="dual triangle" /> 
create dual triangle vertex.
Original triangle vertex come from 
Box15. Please click <input type="button" value="34" /> for example, 
or click <input type="button" value="read" /> to read A,B,C input 
box values at top of drawing board.
Each dual vector has +/- choices.
Three dual vectors have 2*2*2=8 
choices. Select one in 
○1,○2,○3,○4,○5,○6,○7,○8 
<a name="docB_029">&lt;a name="docB_029"&gt;</a>
then click <input type="button" value="fill" />, click to ABC box.
Finally click <input type="button" value="Draw triangle 
ABC sides" /> This draw 
has only dual triangle. If dual
and original can not form "◎" 
which is one inside the other.
Return to
○1,○2,○3,○4,○5,○6,○7,○8 
try another one.
2011-07-06-20-36 here

<a name="docB_030">&lt;a name="docB_030"&gt;</a>
[[<a href=#QAbox16>link</a>
Box16, debug ; <input type="button" value="16to15" />
<TEXTAREA id=QAZboxc16 rows=3 cols=50 ></TEXTAREA>
<input type="button" value="35" /> <input type="button" value="verify01" /> <input type="button" value="33MatDet" /> 
Box17, record, apply ; <input type="button" value="record" /> <input type="button" value="apply" /> 
<TEXTAREA id=QAZboxc17 rows=3 cols=50 ></TEXTAREA>
QAboxc17.value='string'
]]
When test <input type="button" value="transpose" /> need <input type="button" value="16to15" /> .
[Box17, record, apply] already 
explained <a href="#docB_016">earlier</a>.
<a name="docB_031">&lt;a name="docB_031"&gt;</a>
<script language="javascript">link1()</script>
<input type="button" value="35" /> and <input type="button" value="verify01" /> is a set.
Original ∆ABC and dual ∆LMN 
have total six vertex vectors.
∆ABC has a,b,c and ∆LMN has 
x,y,z. 
<a name="docB_032">&lt;a name="docB_032"&gt;</a>
When draw semilunar triangle,
a perpendicular to which two 
of x,y,z?
b perpendicular to which two 
of x,y,z?
c perpendicular to which two 
of x,y,z? Not certain.
Copy a,b,c and x,y,z six vectors
to Box16, or click <input type="button" value="35" /> to use
example, then click <input type="button" value="verify01" /> 

<a name="docB_033">&lt;a name="docB_033"&gt;</a>
Answer output to Box17.
Example <input type="button" value="35" /> answer is 
[[
axy=0,0.7841005286907862
axz=0,0
ayz=0.7841005286907862,0
bxy=0.6838238902874513,0
bxz=0.6838238902874513,0
byz=0,0
cxy=0,0
cxz=0,0.7711475494496229
cyz=0,0.7711475494496229
]]
<a name="docB_034">&lt;a name="docB_034"&gt;</a>
axz=0,0 say: take a,x,z to draw 
triangle.
byz=0,0 say: take b,y,z to draw 
triangle.
cxy=0,0 say: take c,x,y to draw 
triangle.
They are three semilunar triangles
<a href=tute0055.htm#QCDraw>
QCDraw fig.6</a> has six 90 deg. sides.
They are created by [verify01]
<input type="button" value="verify01" /> is a drawing tool.
2011-07-06-20-53 stop

<a name="docB_035">&lt;a name="docB_035"&gt;</a>
2011-07-07-16-47 start
Just created function yearMdhm(). 
Output in Box11 added one line
/***** Data created on 2011-07-10-10-45-17;Sun *****/
for 
year-month-day-hour-minute-second
and weekday name.
Time stamp is important to every
user.
2011-07-07-16-50 stop

<a name="docB_036">&lt;a name="docB_036"&gt;</a>
<script language="javascript">link1()</script>
2011-07-09-18-18 start
During work, find mat33det()
which is not used. Build one
click button <input type="button" value="33MatDet" /> 
to use it. 
Please store a 3 by 3 matrix to
Box16. Click button. matrix 
determinant value output to 
Box17. All matrix in 
"Box15, rotation matrix" are
rotation matrix. Use <input type="button" value="33MatDet" /> 
to verify. 
All rotation matrix has 
determinant value one
2011-07-09-18-25 stop
</font></pre>





















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local
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Save graph code to same folder as htm files.
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<!--a0071047-->
File name sphtrian.htm mean
<br>
SPHerical TRIANgle .htm
<br>
Chinese version is sphtriac.htm

<br>


<br>

This file:Spherical triangle graph page
<br>
<a href=http://freeman2.com/sphtrian.htm>
http://freeman2.com/sphtrian.htm
</a>
<br>
First upload 2011-07-10 
<br>

<br>
Thank you for visiting Freeman's page.　 
<br>
Freeman Liu, HsinHan 劉鑫漢　
<br>
2011-07-07-12-15

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Use Rodrigues Construction to prove 
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2011-06-14-14-40 create
study0008.htm 

2011-06-28-08-56
copy stud0008.htm tutc0055.htm

2011-06-30-09-37
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Spherical Triangle graph page

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