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¯u¥¿ªº¤G¦¸¨ç¼Æ¡^³o¬O¬õ½uªºÁͶաA
¬õ½u¡Ra=1, b=1, c=1, d=-0.1
·í d ¬°¦³­­­È¡A¦P®É a Áͪñ©ó¹s®É¡A¤½¦¡Áͪñ©ó
1/x ¦±½u¡A³o¬OÂŽuªºÁͶաA
ÂŽu¡Ra=0.762, b=-0.686, c=-3.105, d=-1.143
9711061030 ¦¹

    109,110­¶

<a name=a608>
9711061033 ¦¹
¶ë©Z½u Serpentine (¤û¹y 1701)
x*x*y + a*a*y - b*b*x = 0
y = b*b*x/(x*x + a*a)
¦¹¦¡ªº·¥§¤¼Ð¤½¦¡¬°
r*r*cos(theta)*cos(theta)=b*b**cot(theta)-a*a
9711061035 ¦¹

    111,112­¶



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<a name=a609>
9711061539©l
¤Ú´µ¸¯¤G¦¸¦±½u Limacon of Pascal (Pascal 1650)
x=cos(t)*(2*a*cos(t)+b)
y=sin(t)*(2*a*cos(t)+b)
-PI<=t<=+PI a=b=1, 1.5, 2, 2.5, 3
·í a ¤£µ¥©ó b ®É¡A¦³¤£¦Pªº¦±½u¡AŪªÌ¥i¥HÂIÀ»
¡u­×§ï 609¡v¡A§ïÅܰѼƭȡA¦b¶À½u°Ï¤º¡AÂIÀ»¡uµe¹Ï¡v
§ó·s¹Ï®×¡C 9711061554¦¹


    113,115­¶

<a name=a610>
9711071703 ©l
¬_´µ°¨¦±½u Cardioid (Koersma 1689)
°Ñ¼Æ¤½¦¡¬°
x=2*a*cos(t)*(1+cos(t))
y=2*a*sin(t)*(1+cos(t))
-PI<=t<=+PI a=1,2,3,4,5;
·¥§¤¼Ð¤½¦¡¬°
r=2*a*(1+cos(t))
ª½¨¤§¤¼Ð¤½¦¡¬°
(x*x+y*y-2*a*x)^2 = 4*a*a*(x*x+y*y)
9711071708 ¤î

    118,120­¶

<a name=a611>
9711061610¦¹
ª×¤K¦±½u Eight Curve, Lemniscate of Gerono
ª½¨¤§¤¼Ð¤½¦¡¬°¡Rx*x*x*x = a*a*(x*x - y*y)
·¥§¤¼Ð¤½¦¡¬°¡Rr*r = a*a*[sec(theta)]^4*cos(2*theta)
°Ñ¼Æ¤½¦¡¬°¡Rx = a*cos(t)
¤Î y = a*sin(t)*cos(t)
-PI<=t<=+PI a=2,4,6,8,10
9711061614¦¹

    124,125­¶

<a name=a612>¡@¥Ø¿ý
9711061611 ¦¹
§ô¸y¦±½u Bullet Nose Schoute 1885
¦bÂù¦±½u x*x/a/a - y*y/b/b = 1 ¤W¥ô·NÂI P
µe¤@±ø¤Á½u¡A¤Á½u»P x ¶b¤Î y ¶b¬Û¥æ©ó¨âÂI¡C
¦b¤Á½u»P x ¶bªº¥æÂIµe¤@±ø¹ï y ¶bªº¥­¦æ½u¡A
¦b¤Á½u»P y ¶bªº¥æÂIµe¤@±ø¹ï x ¶bªº¥­¦æ½u¡A
¨â±ø¥­¦æ½uªº¥æÂI¬° Q¡A¥ô·NÂI P ²¾°Ê®É¡A Q ÂI
ªº­y¸ñ¬O§ô¸y¦±½u¡C
§ô¸y¦±½uªº¤½¦¡¬°
a*a/x/x - b*b/y/y = 1
§ô¸y¦±½uªº°Ñ¼Æ¤½¦¡¬°
x=a*cos(t)
y=b/tan(t) ¡e¤£¯à¥Î y=b*cot(t) 9711031656¡f
-PI<=t<=+PI a=2,4,6,8,10; b=4
9711061629 ¦¹

    126,129­¶

<a name=a613>
9711061631 ¦¹
¤Q¦r¦±½u Cross curve
¦b¾ò¶ê x*x/a/a + y*y/b/b = 1 ¤W¥ô·NÂI P
µe¤@±ø¤Á½u¡A¤Á½u»P x ¶b¤Î y ¶b¬Û¥æ©ó¨âÂI¡C
¦b¤Á½u»P x ¶bªº¥æÂIµe¤@±ø¹ï y ¶bªº¥­¦æ½u¡A
¦b¤Á½u»P y ¶bªº¥æÂIµe¤@±ø¹ï x ¶bªº¥­¦æ½u¡A
¨â±ø¥­¦æ½uªº¥æÂI¬° Q¡A¥ô·NÂI P ²¾°Ê®É¡A Q ÂI
ªº­y¸ñ¬O¤Q¦r¦±½u¡C
¤Q¦r¦±½uªº¤½¦¡¬°
a*a/x/x + b*b/y/y = 1
¤Q¦r¦±½uªº°Ñ¼Æ¤½¦¡¬°
x=a*sec(t)=a/cos(t)
y=b*csc(t)=b/sin(t)
-PI<=t<=+PI a=b = 2,4,6,8,10
9711061634 ¦¹


    127,131­¶

<a name=a614>
9711061635 ¦¹
¤T¦y¦±½u Deltoid (Euler 1745) (tricuspid)
¤@­Ó¥b®|¬° a ªº¶ê½L¦b¥b®|¬° 3*a ªº¶ê¬}¤ººu°Ê¡A¶ê½L
Ãäªu¤W¤@ÂIªº­y¸ñ¡A§Î¦¨¤T¦y¦±½u¡C°Ñ¼Æ¤½¦¡¦p¤U
x=a*(2*cos(t)+cos(2*t))
y=a*(2*sin(t)-sin(2*t))
-PI<=t<=+PI ; a=1,2,3
ª½¨¤§¤¼Ð¤½¦¡¤ñ¸û½ÆÂø¡A¦p¤U
(x*x+y*y)^2 - 8*a*x*(x*x-3*y*y)
+18*a*a*(x*x+y*y) = 27*a*a*a*a
9711061641 ¦¹


    131,135­¶

<a name=a615>¡@¥Ø¿ý
9711061648 ¦¹
¥§¬_¦Ì¼w¦±½u Conchoid of Nicomedes (Nicomedes, 225 B.C.)
x=b+a*cos(t)
y=tan(t)*(b+a*cos(t))
a=2,4,6,8; b=2
¥|±ø¦±½u³£¥H x=2 ªººñ¦â²Êª½½u¬°¥D¨¤¡C½Ò¥»
¥|¤Q¤E­¶»¡¡u¨©´ß½u¡v (conchoid) ªº©w¸q»Ý­n
¦³¤@±ø¦±½u¡A¤Î¨â­Ó©T©wÂI¡C¦ý¬O¡A½Ò¥»¤@¤T¤C­¶ªº
°Q½×¥u¦³¤@­Ó©T©wÂI (0,0) ¡A¥t¥~¤@­Ó©T©wÂI
¦b¨ºùØ¡H¨S¦³¥æ¥N¡C¡]¦±½u¬O x=2 ªººñ¦â²Êª½½u¡^
©Î³\´X¦ó±M·~ªºÅªªÌª¾¹Dµª®×¡C 9711061712¦¹

    135,139­¶

<a name=a616>
9711061832 ©l
¤×¦h´µ¦±½u Kampyle of Eudoxus
ª½¨¤§¤¼Ð¨ç¼Æ¬° x*x*x*x = a*a*(x*x+y*y)
·¥§¤¼Ð¨ç¼Æ¬° r*cos(theta)*cos(theta)=a
°Ñ¼Æ¤½¦¡¬° x = a/cos(t) = a*sec(t)
y = a*sin(t)/cos(t)/cos(t) = a*sec(t)*tan(t)
-PI/2<t<3*PI/2; a=2,4,6,8
9711061836 ¦¹

    141,143­¶


<a name=b003>
9711061848¦¹ (144,145 ­¶)
¥èªi¼w¦±½u Hippopede (Proclus 75 B.C.)
µe¹Ï²Ä¤T¨÷ http://freeman2.com/graph03c.htm
¦³¥èªi¼w¦±½u¡C¨ú¾\¤èªk¬°¥´¶} graph03c.htm
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­È¬°¥èªi¼w¦±½u¡A¤£­nÂIÀ»¡u¨ú¥Î¡v¡Aª½±µÂIÀ»¡uµe½u¡v¡A
Àò±o¥èªi¼w¦±½u¡C¡]¦pªGÂIÀ»¡u¨ú¥Î¡v¶s¡Aªì©l¤º©w­È®ø
¥¢¡AµLªk¥s¦^¡A¥u¦³­«·s¶}±Òºô­¶¡A±o¨ìªì©l¤º©w­È¡^
¥èªi¼w¦±½uªº°Ñ¼Æ¤½¦¡¬°
x=2*cos(t)*sqrt(a*b-b*b*sin(t)*sin(t))
y=2*sin(t)*sqrt(a*b-b*b*sin(t)*sin(t))
-PI<=t<=+PI ;
graph03c.htm ªº¥èªi¼w¦±½u¡A¨Ï¥Î±`¼Æ­È¦p¤U
a=2,3,4,5,6; b=4 ¤£ÅÜ¡C

<a name=a617>
¥»¨÷ tutc0004.htm ªº¥èªi¼w¦±½u¡A¨Ï¥Î±`¼Æ­È¦p¤U
a=1 ¤£ÅÜ¡C b=1,2,3,4
¥|±ø¦±½uªu y ¶b¤è¦V¬G·N¥­²¾¡AÁ×§K­«Å|¤£²M¡C
¥èªi¼w¦±½uªºª½¨¤§¤¼Ð¤½¦¡¬°
(x*x+y*y)*(x*x+y*y)+4*b*(b-a)*(x*x+y*y)=4*b*b*x*x
¥èªi¼w¦±½uªº·¥§¤¼Ð¤½¦¡¬°
r*r = 4*b*(a-b*sin(theta)*sin(theta))
9711061904¦¹


    144,145­¶

<a name=a618>¡@¥Ø¿ý
9711061913 ¦¹
¶w³»¦±½u Bicorn (Sylvester 1864)
¶w³»¦±½uªºª½¨¤§¤¼Ð¤½¦¡¬°
(x*x+2*a*y-a*a)^2 = y*y*(a*a-x*x)
¶w³»¦±½uªº°Ñ¼Æ§¤¼Ð¤½¦¡¬°
x=a*sin(t)
y=a*cos(t)*cos(t)*(2+cos(t))/(3+sin(t)*sin(t))
-PI<=t<=+PI ; a=2,4,6,8
9711061918 ¦¹


    147,148­¶

<a name=a619>
9711061922 ¦¹
±ù§Î¦±½u Piriform (De Longchamps, 1886)
¤SºÙ¬° pear-shaped quartic.
±ù§Î¦±½uªº°Ñ¼Æ¤½¦¡¬°
x=a*(1+sin(t))
y=b*cos(t)*(1+sin(t))
a=1,2,3,4; b=4
-PI/2<=t<=+3*PI/2
±ù§Î¦±½uªºª½¨¤§¤¼Ð¤½¦¡¬°
a*a*a*a*y*y = b*b*x*x*x*(2*a-x)
±ù§Î¦±½uªº·¥§¤¼Ð¤½¦¡¬°
b*b*r*r*(cos(theta))^4-2*a*b*b*r*(cos(theta))^3
+a*a*a*a*sin(theta)*sin(theta) = 0
9711061930 ¦¹


    148,149­¶


<a name=b004>
9711070816©l
µe¹Ï²Ä¤T¨÷ http://freeman2.com/graph03c.htm
¦³¡m¯S®í¥­­±¦±½u¥Ø¿ý¡n151­¶¡A°­Áy¦±½u Devil's curve.
Cramer, 1750. 152­¶¤W¥b­¶¬°°­Áy¦±½u¹Ï§Î¡C
°­Áy¦±½u¥Î¨â­Ó±`¼Æ a , b
ª½¨¤§¤¼Ð¤½¦¡¬° y*y*y*y - a*a*y*y = x*x*x*x - b*b*x*x
·¥§¤¼Ð¤½¦¡
r*r*(sin(t)^2-cos(t)^2)=a*a*sin(t)^2-b*b*cos(t)^2
°Ñ¼Æ¤½¦¡¬°
x(t)=cos(t)*sqrt((a*a*sin(t)*sin(t)-b*b*cos(t)*cos(t))/(sin(t)*sin(t)-cos(t)*cos(t)))
y(t)=sin(t)*sqrt((a*a*sin(t)*sin(t)-b*b*cos(t)*cos(t))/(sin(t)*sin(t)-cos(t)*cos(t)))
°Ñ¼Æ t ªº½d³ò¬O
-PI/4 < t < PI/4 ; 3*PI/4 < t < 5*PI/4
¥O u=atan(b/a) ¡A °Ñ¼Æ t ªº½d³ò¬O
u <= t <= PI-u ; -(PI-u) <= t <= -u
½Ò¥» 152 ­¶¤W¥b­¶¬°°­Áy¦±½u¹Ï§Î¡A¤¤¶¡¬O¤@­Óª½¥ßªº¡u8¡v
¦ý¬O¡Aµe¹Ï²Ä¤T¨÷ graph03c.htm ¤¤¶¡¬O¨â­Ó¾îª×ªº¡u8¡v¡A
¤£ª¾¹D¦ó³B²£¥Í®t²§¡H¨ú¾\¹Ï§Î¤èªk¬O¥´¶}
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¼Æ¦r¡u9¡v¡AÂIÀ»¡u¨ú¥Î¡v¡A¦¹®É¥i¥H­×§ï¦±½u°Ñ¼Æ¡Aº¡·N¤§«á¡A¦A
ÂIÀ»¡uµe½u¡v¡C¥»¨÷ tutc0004.htm ¤£¦A­«´_¡C
9711070838 ¦¹



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¡m¯S®í¥­­±¦±½u¥Ø¿ý¡n²Ä¤»³¹¡R¡u°ª¦¸¥N¼Æ¦±½u¡v
²Ä¤@¤»¹s­¶¦Ü¤@¤K¤T­¶¡C


<a name=a620>
9711070927©l
¥~±Û½ü½u Epitrochoid (Dűrer, 1525)
¥O S ¬°°Ê¶ê¡A¥b®|¬° b ¡F¥O C ¬°¦a¶ê¡A¥b®|¬° a
°Ê¶ê¦b¦a¶ê¥~­±ºu°Ê¡A°Ê¶ê¤W¦³¤@ÂI p ¶ZÂ÷°Ê¶ê h
¥O m=a+b ¬°°Ê¶ê¤ß¦Ü¦a¶ê¤ßªº¶ZÂ÷¡A p ÂIªº­y¸ñ
x=m*cos(t)-h*cos(m*t/b)
y=m*sin(t)-h*sin(m*t/b)
m=4,6,8,10,18; b=2
³oùا⤭±i¹Ï©ñ¦b¤@°_¡A¤Q¤À¾ÖÀ½¡A½ÐÂIÀ»¡u­×§ï 620¡v
¦b¡u§R°£¨ç¼Æ¡v¥k°¼§R°£´X±ø¦±½u¡]¨Ò¦p§R°£¤G¡B¤T¡B¥|¡^¡A
¦b¶À½u°Ï¤ºÂIÀ»¡uµe¹Ï¡v¡Aµe¤@±ø¦±½u¡Cµe²Ä¤G±ø®É¡AÂI
À»¡u¨ú¥Î­ìÃD¡v¡A¦A§R°£¤£­nªº¦±½u¡C
9711070956


    160,164­¶

<a name=a621>¡@¥Ø¿ý
9711071040©l
¤º±Û½ü½u Hypotrochoid
¥O S ¬°°Ê¶ê¡A¥b®|¬° b ¡F¥O C ¬°¦a¶ê¡A¥b®|¬° a
°Ê¶ê¦b¦a¶ê¤º­±ºu°Ê¡A°Ê¶ê¤W¦³¤@ÂI p ¶ZÂ÷°Ê¶ê h
¥O n=a-b ¬°°Ê¶ê¤ß¦Ü¦a¶ê¤ßªº¶ZÂ÷¡A p ÂIªº­y¸ñ
x=n*cos(t)+h*cos(n*t/b)
y=n*sin(t)-h*sin(n*t/b)
n=2,4,6,14; b=2
³oùاâ¥|±i¹Ï©ñ¦b¤@°_¡A¤Q¤À¾ÖÀ½¡A½ÐÂIÀ»¡u­×§ï 621¡v
¦b¡u§R°£¨ç¼Æ¡v¥k°¼§R°£´X±ø¦±½u¡]¨Ò¦p§R°£¤G¡B¤T¡B¥|¡^¡A
¦b¶À½u°Ï¤ºÂIÀ»¡uµe¹Ï¡v¡Aµe¤@±ø¦±½u¡Cµe²Ä¤G±ø®É¡AÂI
À»¡u¨ú¥Î­ìÃD¡v¡A¦A§R°£¤£­nªº¦±½u¡C
9711071048¤î


    165,168­¶

<a name=a622>
9711071050©l
´f®Ú´µ¥~±Û½ü½u Nephroid (Huygens, 1678)
´f®Ú´µ¥~±Û½ü½u¬O¨â­Ó¦yÂIªº¥~±Û½ü½u¡A¤½¦¡¬°
x=a*(3*cos(t)-cos(3*t))
y=a*(3*sin(t)-sin(3*t))
-PI<=t<=+PI ; a=1,2,3,4
ª½¨¤§¤¼Ð¤½¦¡¬°
(x*x+y*y-4*a*a)^3 = 108*a*a*a*a*y*y
·¥§¤¼Ð¤½¦¡¬°
(r/2/a)^(2/3) = (sin(theta/2))^(2/3)+(cos(theta/2))^(2/3)
9711071054¤î


    169,171­¶

<a name=a623>
9711071056©l
ÃMÃ䤺±Û½ü½u Hypocycloid ¡C p ÂI ¦b°Ê¶êÃä¤W h=b
¥O S ¬°°Ê¶ê¡A¥b®|¬° b ¡F¥O C ¬°¦a¶ê¡A¥b®|¬° a
°Ê¶ê¦b¦a¶ê¤º­±ºu°Ê¡A°Ê¶ê¤W¦³¤@ÂI p ¶ZÂ÷°Ê¶ê h
¥O n=a-b ¬°°Ê¶ê¤ß¦Ü¦a¶ê¤ßªº¶ZÂ÷¡A p ÂIªº­y¸ñ
x=n*cos(t)+b*cos(n*t/b)
y=n*sin(t)-b*sin(n*t/b)
¤W¦¡¤w¸g¥N¤J¤F h=b ¡C b=2, n=2,4,6,8,10
b=2, n=2 °h¤Æ¬°¤@±ø¦b x ¶b¤Wªºª½½u¡C
9711071100¦¹


    171,173­¶

<a name=a624>¡@¥Ø¿ý
9711071101¦¹
¬P¥ú¦±½u Astroid (Roemer, 1674; Bernoulli, 1691)
¬P¥ú¦±½u¬O¥|­Ó¦yÂIªº¤º±Û½ü½u
x=a*(3*cos(t)+cos(3*t))
y=a*(3*sin(t)-sin(3*t))
-PI<=t<=+PI ; a=1,2,3,4
¤½¦¡¤]¥i¥H¼g¬°
x=4*a*cos(t)*cos(t)*cos(t)
y=4*a*sin(t)*sin(t)*sin(t)
ª½¨¤§¤¼Ð¤½¦¡¬°
x^(2/3) + y^(2/3) = a^(2/3)
®i¶}¤§«á¡AÅܬ°
(x*x+y*y-a*a)^3 + 27*a*a*x*x*y*y = 0
9711071105¦¹


    172,174­¶


<a name=b005>
9711071109
µe¹Ï²Ä¤T¨÷ http://freeman2.com/graph03c.htm
[¯S®í¥­­±¦±½u¥Ø¿ý] ISBN 0-486-60288-5, 175,177­¶
ª´ºÀ¦±½u Rhodonea. Grandi, 1723. ¬O¥~Â\½u¹ï¤¤¤ß
¤§««¨¬½u¡C·¥§¤¼Ð¤½¦¡¬° r=a*cos(b*theta)
°Ñ¼Æ¤½¦¡¬°
x(t)=a*cos(b*t)*cos(t)
y(t)=a*cos(b*t)*sin(t)
­Y b ¬O©_¼Æ¡A«h¦³ b ­Óªáä¡C­Y b ¬O°¸¼Æ¡A«h¦³
2*b ­Óªáä¡C¯S¨Ò¬° b=3,2,1/3 9506221714
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¦±½u°Ñ¼Æ¡Aº¡·N¤§«á¡A¦A ÂIÀ»¡uµe½u¡v¡C
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9711071122¦¹
µe¹Ï²Ä¤T¨÷ http://freeman2.com/graph03c.htm
[¯S®í¥­­±¦±½u¥Ø¿ý] ISBN 0-486-60288-5, 178,183­¶
¥]­}©}¦±½u. Bowditch, 1815. §O¦W Lissajous ¦±½u¡C
­Y n ¬O¦³²z¼Æ¡A¦±½u¬°¥N¼Æ¦±½u¡A§_«h¬°¶W¶V¦±½u¡C
¦±½u§¹¥þ§½­­©ó |x|<=a ¤Î |y|<=b ½d³ò¤º¡C
°Ñ¼Æ¤½¦¡¬°
x(t)=a*sin(n*t+d)=9*sin(0.8*t+3.14159)
y(t)=b*sin(t)=8*sin(t)
­Y b ¬O©_¼Æ¡A«h¦³ b ­Óªáä¡C­Y b ¬O°¸¼Æ¡A«h¦³
2*b ­Óªáä¡C¯S¨Ò¬° b=3,2,1/3       9506221714
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9711071127¦¹



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<a name=b007>
9711071307 ©l ½Ð±z°Ê¤â«Ø¥ß¦±½u¹Ï
µe¹Ï²Ä¤T¨÷ http://freeman2.com/graph03c.htm
[¯S®í¥­­±¦±½u¥Ø¿ý] ISBN 0-486-60288-5 184,185­¶
¶}¤èµo´²Á³±Û½u¡C½Ò¥»¤½¦¡ r=exp(a*theta)
graph03c.htm §ï¬°
r=sqrt(theta) ©Î r=sqrt(t)
¦]¬° x=r*cos(t) ¤Î y=r*sin(t)
¹ê»Ú«ü¥O¨Ï¥Î x(t)=sqrt(t)*cos(t);
¤Î y(t)=sqrt(t)*sin(t);
r=exp(theta) ¥b®|¨³³tµo´²¡A r=sqrt(t) ¥b®|ºC³tµo´²
©Ò¥H¡A graph03c.htm ²Ä¤G¸¹¦±½u r=sqrt(t) ¹ê»Ú¤W
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x = a*t - h*sin(t)
y = a - h*cos(t)
­Y P ÂI¦b¥b®|¤W h=a ¬õ¦âªuÃä±Û½ü½u ordinary cycloid
­Y P ÂI¦b¥b®|¤º h<a ÂŦ⤺Ãä±Û½ü½u curtate cycloid
­Y P ÂI¦b¥b®|¥~ h>a ¶Â¦â¥~Ãä±Û½ü½u prolate cycloid
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(Hippias of Elis, 430 B.C.)
y = x*cot(PI*x/2/a)
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y=a*(0.5*(exp(x/a)+exp(-x/a)))
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x = a*ln(1/cos(t) + tan(t)) - a*sin(t)
y = a*cos(t)
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A Catalog of Special Plane Curves.
§@ªÌ J Dennis Lawrence , 
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­ì§@ªÌ¡R Ken Kikuchi Comment me: kikuchi@mix.or.jp
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stepf(t,-2)*(1-stepf(t,2)) ¬° f1(t)=-3.09*sin(t) ¶}±Ò [-2,2]
-3.09728632250*stepf(t,-2)*(1-stepf(t,2))*Math.sin(t)
stepf(t,2)*(1-stepf(t,5))¬°f2(t)=exp(-t+PI)*sin(t-PI)¶}[2,5]
+stepf(t,2)*(1-stepf(t,5))*Math.exp(-(t-PI))*Math.sin(t-PI)
stepf(t,5)*(1-stepf(t,8)) ¬° f3(t)=x*x*x/108.75-1 ¶}±Ò [5,8]
+stepf(t,5)*(1-stepf(t,8))*(x*x*x/108.75-1)
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