School of Mathematics and Statistics, University of St Andrews, Scotland
による数学史に関する膨大な資料のページの中の次のページに挙げられている曲線を
Mathematicaにより描いたものである
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html

<< Graphics`Graphics` << Graphics`ImplicitPlot`

Astroid

ImplicitPlot[(x^2)^(1/3) + (y^2)^(1/3) == 1, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_3.gif]

Bicorn

ImplicitPlot[y^2 (1 - x^2) == (x^2 + 2 y - 1)^2, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_5.gif]

Cardioid

ImplicitPlot[(x^2 + y^2 - 2 x)^2 == 4 (x^2 + y^2), {x, -1, 4}] ;

[Graphics:HTMLFiles/FamousCurves_7.gif]

Cartesian Oval

m = 1/2 ; a = 1 ; c = 3 ; ImplicitPlot[((1 - m^2) (x^2 + y^2) + 2 m^2 c x + a^2 - m^2 c^2)^2 == 4 a^2 (x^2 + y^2), {x, -6, 4}] ;

[Graphics:HTMLFiles/FamousCurves_9.gif]

m = 1/3 ; a = 1 ; c = 2 ; ImplicitPlot[((1 - m^2) (x^2 + y^2) + 2 m^2 c x + a^2 - m^2 c^2)^2 == 4 a^2 (x^2 + y^2), {x, -6, 4}] ;

[Graphics:HTMLFiles/FamousCurves_11.gif]

Casinian Ovals

ImplicitPlot[(x^2 + y^2)^2 - 2 (x^2 - y^2) == -1/4, {x, -4, 4}] ;

[Graphics:HTMLFiles/FamousCurves_13.gif]

ImplicitPlot[(x^2 + y^2)^2 - 2 (x^2 - y^2) == 1/4, {x, -4, 4}] ;

[Graphics:HTMLFiles/FamousCurves_15.gif]

Catenary

Plot[Cosh[x], {x, -3, 3}] ;

[Graphics:HTMLFiles/FamousCurves_17.gif]

Caylely's Sextic

ImplicitPlot(4 (x^2 - x + y^2)^3 == 27 (x^2 + y^2)^2, {x, -1, 5}, PlotRange -> All) ;

[Graphics:HTMLFiles/FamousCurves_19.gif]

Circle

ImplicitPlot[x^2 + y^2 == 1, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_21.gif]

Cissoid  of  Diocles

ImplicitPlot[y^2 == x^3/(2 - x), {x, 0, 1.6}] ;

[Graphics:HTMLFiles/FamousCurves_23.gif]

Cochleoid

PolarPlot[Sin[t]/t, {t, 0, 20}, PlotRange -> All] ;

[Graphics:HTMLFiles/FamousCurves_25.gif]

Conchoid

ImplicitPlot[(x - 1/3)^2 (x^2 + y^2) == x^2, {x, -1, 2}, PlotRange -> {-3, 3}] ;

[Graphics:HTMLFiles/FamousCurves_27.gif]

Cycloid

a = 1 ; h = 1 ; ParametricPlot[{a t - h Sin[t], a - h Cos[t]}, {t, 0, 4 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_29.gif]

Devil's  Curve

ImplicitPlot[y^4 - x^4 - 24 y^2 + 25 x^2 == 0, {x, -7, 7}] ;

[Graphics:HTMLFiles/FamousCurves_31.gif]

Double Folium

ImplicitPlot[(x^2 + y^2)^2 == 4 x y^2, {x, -2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_33.gif]

Durer's Shell Curves

b = 3 ; ImplicitPlot[(x^2 + x y + x - b^2)^2 == (b^2 - x^2) (x - y + 1)^2, {x, -b, b}, PlotRange -> {-5, 7}] ;

[Graphics:HTMLFiles/FamousCurves_35.gif]

ImplicitPlot[(x^2 + x y + x - b^2)^2 == (b^2 - x^2) (x - y + 1)^2, {x, -b, b}, PlotRange -> {-5, 5}] ;

[Graphics:HTMLFiles/FamousCurves_37.gif]

ImplicitPlot[(x^2 + x y + x - b^2)^2 == (b^2 - x^2) (x - y + 1)^2, {x, -b, b}, PlotRange -> {-3, 3}] ;

[Graphics:HTMLFiles/FamousCurves_39.gif]

Figure Eight Curve

ImplicitPlot[x^4 == x^2 - y^2, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_41.gif]

Eloipse

ImplicitPlot[x^2 + 2^2 y^2 == 1, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_43.gif]

Epicycloid

a = 8 ; b = 5 ; ParametricPlot[{(a + b) Cos[t] - b Cos[(a/b + 1) t], (a + b) Sin[t] - b Sin[(a/b + 1) t]}, {t, 0, 10 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_45.gif]

a = 16 ; b = 5 ; ParametricPlot[{(a + b) Cos[t] - b Cos[(a/b + 1) t], (a + b) Sin[t] - b Sin[(a/b + 1) t]}, {t, 0, 10 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_47.gif]

a = 24 ; b = 5 ; ParametricPlot[{(a + b) Cos[t] - b Cos[(a/b + 1) t], (a + b) Sin[t] - b Sin[(a/b + 1) t]}, {t, 0, 10 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_49.gif]

Epitrochoid

a = 5 ; b = 3 ; c = 5 ; ParametricPlot[{(a + b) Cos[t] - c Cos[(a/b + 1) t], (a + b) Sin[t] - c Sin[(a/b + 1) t]}, {t, 0, 6 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_51.gif]

a = 5 ; b = 3 ; c = 3 ; ParametricPlot[{(a + b) Cos[t] - c Cos[(a/b + 1) t], (a + b) Sin[t] - c Sin[(a/b + 1) t]}, {t, 0, 6 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_53.gif]

a = 5 ; b = 3 ; c = 4 ; ParametricPlot[{(a + b) Cos[t] - c Cos[(a/b + 1) t], (a + b) Sin[t] - c Sin[(a/b + 1) t]}, {t, 0, 6 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_55.gif]

Equiangular Spiral

b = (7 π)/16 ; PolarPlot[e^(t Cot[b]), {t, 0, 8 π}] ;

[Graphics:HTMLFiles/FamousCurves_57.gif]

b = (8 π)/17 ; PolarPlot[e^(t Cot[b]), {t, 0, 8 π}] ;

[Graphics:HTMLFiles/FamousCurves_59.gif]

Fermat's Spiral

PolarPlot[{t^(1/2), -t^(1/2)}, {t, 0, 8 π}] ;

[Graphics:HTMLFiles/FamousCurves_61.gif]

Folium

a = 1 ; b = 2 a ; ImplicitPlot[(x^2 + y^2) (y^2 + x (x + b)) == 4 a x y^2, {x, -2 a, 2 a}] ;

[Graphics:HTMLFiles/FamousCurves_63.gif]

a = 1 ; b = a ; ImplicitPlot[(x^2 + y^2) (y^2 + x (x + b)) == 4 a x y^2, {x, -2 a, 2 a}] ;

[Graphics:HTMLFiles/FamousCurves_65.gif]

a = 1 ; b = 0 ; ImplicitPlot[(x^2 + y^2) (y^2 + x (x + b)) == 4 a x y^2, {x, -2 a, 2 a}] ;

[Graphics:HTMLFiles/FamousCurves_67.gif]

a = 1 ; b = 4 a ; ImplicitPlot[(x^2 + y^2) (y^2 + x (x + b)) == 4 a x y^2, {x, -2 a, 2 a}] ;

[Graphics:HTMLFiles/FamousCurves_69.gif]

Folium of Descartes

ImplicitPlot[x^3 + y^3 == 3 x y, {x, -2, 2}, PlotRange -> {-2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_71.gif]

Frfeeth's Nephroid

PolarPlot[1 + 2 Sin[t/2], {t, 0, 4 π}] ;

[Graphics:HTMLFiles/FamousCurves_73.gif]

Frequency Curve

Plot[(2 π)^(1/2) e^(-x^2/2), {x, -4, 4}] ;

[Graphics:HTMLFiles/FamousCurves_75.gif]

Hyperbola

ImplicitPlot[x^2 - y^2 == 1, {x, -4, 4}] ;

[Graphics:HTMLFiles/FamousCurves_77.gif]

Hyperbolic Spiral

PolarPlot[1/t, {t, 1, 30}, PlotRange -> All] ;

[Graphics:HTMLFiles/FamousCurves_79.gif]

Hypocycloid

a = 5 ; b = 3 ; ParametricPlot[{(a - b) Cos[t] + b Cos[(a/b - 1) t], (a - b) Sin[t] - b Sin[(a/b - 1) t]}, {t, 0, 6 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_81.gif]

Hypotrochoid

a = 5 ; b = 7 ; c = 2.2 ; ParametricPlot[{(a - b) Cos[t] + c Cos[(a/b - 1) t], (a - b) Sin[t] - c Sin[(a/b - 1) t]}, {t, 0, 14 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_83.gif]

Involute of a Circle

ParametricPlot[{Cos[t] + t Sin[t], Sin[t] - t Cos[t]}, {t, 0, 6 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_85.gif]

Kampyle of Eudoxus

ImplicitPlot[x^4 == x^2 + y^2, {x, -2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_87.gif]

Kappa Curve

a = 2 ; b = 1 ; n = 4 ; ImplicitPlot[(x/a)^n + (y/b)^n == 1, {x, -a, a}] ;

[Graphics:HTMLFiles/FamousCurves_89.gif]

Lame Curve

ImplicitPlot[(x^2 + y^2)^2 == x^2 - y^2, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_91.gif]

Lemniscate of Bernoulli

ImplicitPlot[(x^2 + y^2)^2 == x^2 - y^2, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_93.gif]

Limacon of Pascal

a = 1 ; b = 1 ; ImplicitPlot[(x^2 + y^2 - 2 a x)^2 == b^2 (x^2 + y^2), {x, -1, 4 a}] ;

[Graphics:HTMLFiles/FamousCurves_95.gif]

Lissajous Curves

a = 1 ; b = 1 ; c = π/3 ; n = 3 ; ParametricPlot[{a Sin[n t + c], b Sin[t]}, {t, 0, 2 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_97.gif]

Lituus

PolarPlot[{1/t^(1/2), -1/t^(1/2)}, {t, 1, 30}] ;

[Graphics:HTMLFiles/FamousCurves_99.gif]

Neile's Semi-cubical Parabola

ImplicitPlot[y^3 == x^2, {x, -2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_101.gif]

Nephroid

ParametricPlot[{3 Cos[t] - Cos[3 t], 3 Sin[t] - Sin[3 t]}, {t, 0, 2 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_103.gif]

Newton's Diverging Parabolas

a = 1.5 ; b = 1 ; c = 0.95 ; ImplicitPlot[a y^2 == x (x^2 - 2 b x + c), {x, 0, 2}] ;

[Graphics:HTMLFiles/FamousCurves_105.gif]

Parabola

a = 1 ; b = 1 ; c = -2 ; Plot[a x^2 + b x + c, {x, -3, 3}] ;

[Graphics:HTMLFiles/FamousCurves_107.gif]

Pear-shaped Quartic

b = 0.7 ; a = 1 ; ImplicitPlot[b^2 y^2 == x^3 (a - x), {x, 0, a}] ;

[Graphics:HTMLFiles/FamousCurves_109.gif]

Pearls of Sluze

n = 4 ; k = 2 ; a = 4 ; p = 3 ; m = 2 ; ImplicitPlot[y^n == k (a - x)^p x^m, {x, -a, a}] ;

[Graphics:HTMLFiles/FamousCurves_111.gif]

Plateau Curves

m = 5 ; n = 3 ; ParametricPlot[{Sin[(m + n) t]/Sin[(m - n) t], 2 Sin[m t] Sin[n t] Sin[(m - n) t]}, {t, 0, π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_113.gif]

m = 5 ; n = 2 ; ParametricPlot[{Sin[(m + n) t]/Sin[(m - n) t], 2 Sin[m t] Sin[n t] Sin[(m - n) t]}, {t, 0, π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_115.gif]

m = 5 ; n = 4 ; ParametricPlot[{Sin[(m + n) t]/Sin[(m - n) t], 2 Sin[m t] Sin[n t] Sin[(m - n) t]}, {t, 0, π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_117.gif]

m = 5 ; n = 1 ; ParametricPlot[{Sin[(m + n) t]/Sin[(m - n) t], 2 Sin[m t] Sin[n t] Sin[(m - n) t]}, {t, 0, π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_119.gif]

m = 3 ; n = 5 ParametricPlot[{Sin[(m + n) t]/Sin[(m - n) t], 2 Sin[m t] Sin[n t] Sin[(m - n) t]}, {t, 0, π}, AspectRatio -> Automatic] ;

5

[Graphics:HTMLFiles/FamousCurves_122.gif]

Pursuit Curve

c = 1 ; Plot[c x^2 - Log[x], {x, 0, 2}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_124.gif]

Quadratrix of Hippias

Plot[x Cot[(π x)/2], {x, -6, 6}, PlotRange -> {-10, 10}] ;

[Graphics:HTMLFiles/FamousCurves_126.gif]

Rhodenea Curves

k = 5 ; PolarPlot[Sin[k t], {t, 0, π}] ;

[Graphics:HTMLFiles/FamousCurves_128.gif]

Right Strophoid

a = 1 ; Plot[{(x^2 (a - x))/(a + x)^(1/2), -(x^2 (a - x))/(a + x)^(1/2)}, {x, -a, a}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_130.gif]

Serpentine

a = 3 ; b = 2 ; Plot[(a^2 x)/(x^2 + a b), {x, -(a + b), a + b}] ;

[Graphics:HTMLFiles/FamousCurves_132.gif]

Sinusoidal Spirals

p = 2/3 ; PolarPlot[Abs[Cos[p t]]^(1/p), {t, 0, 6 π}] ;

[Graphics:HTMLFiles/FamousCurves_134.gif]

p = 1/3 ; PolarPlot[Abs[Cos[p t]]^(1/p), {t, 0, 6 π}] ;

[Graphics:HTMLFiles/FamousCurves_136.gif]

p = 2/5 ; PolarPlot[Abs[Cos[p t]]^(1/p), {t, 0, 10 π}] ;

[Graphics:HTMLFiles/FamousCurves_138.gif]

Spiral of Archimedes

PolarPlot[t, {t, 0, 10 π}] ;

[Graphics:HTMLFiles/FamousCurves_140.gif]

Spiric Sections

r = 2.1 ; a = 1 ; c = 1 ; ImplicitPlot[(r^2 - a^2 + c^2 + x^2 + y^2)^2 == 4 r^2 (x^2 + c^2), {x, -(a + c + r), a + c + r}] ;

[Graphics:HTMLFiles/FamousCurves_142.gif]

Straight Line

a = 1 ; b = -1 ; c = 2 ; d = 3 ; ParametricPlot[{a t + b, c t + d}, {t, -2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_144.gif]

Talbot's Curve

a = 1 ; b = 1 ; f = 0.95 ; ParametricPlot[{((a^2 + f^2 Sin[t]^2) Cos[t])/a, ((a^2 - 2 f^2 + f^2 Sin[t]^2) Sin[t])/b}, {t, 0, 2 π}, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_146.gif]

Tractrix

ParametricPlot[{1/Cosh[t], t - Tanh[t]}, {t, -2, 2}, PlotRange -> All, AspectRatio -> Automatic] ;

[Graphics:HTMLFiles/FamousCurves_148.gif]

Tricuspoid

ImplicitPlot[(x^2 + y^2 + 12 x + 9)^2 == 4 (2 x + 3)^3, {x, -2, 3}] ;

[Graphics:HTMLFiles/FamousCurves_150.gif]

Trident of Newton

c = 1 ; d = 1 ; e = -1/2 ; f = -1/20 ; Plot[c (x - d)^2 + e + f/x, {x, -2, 2}, AspectRatio -> Automatic, PlotRange -> {-2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_152.gif]

Trifolium

ImplicitPlot[(x^2 + y^2) (y^2 + x (x + 1)) == 4 x y^2, {x, -1, 1}] ;

[Graphics:HTMLFiles/FamousCurves_154.gif]

Trisectrix of Maclaurin

ImplicitPlot[y^2 (1 + x) == x^2 (3 - x), {x, -1, 4}, PlotRange -> {-3, 3}] ;

[Graphics:HTMLFiles/FamousCurves_156.gif]

Tschirnhaus's Cubic

a = 1 ; ImplicitPlot[3 a y^2 == x (x - a)^2, {x, 0, 2 a}] ;

[Graphics:HTMLFiles/FamousCurves_158.gif]

Watt's Curve

a = 2.2 ; b = 6.4 ; c = 4 ; PolarPlot[{(b^2 - (a Sin[t] + (c^2 - a^2 Cos[t]^2)^(1/2))^2)^(1/2), (b^2 - (a Sin[t] - (c^2 - a^2 Cos[t]^2)^(1/2))^2)^(1/2)}, {t, 0, 2 π}] ;

[Graphics:HTMLFiles/FamousCurves_160.gif]

Witch of Agnesi

Plot[1/(x^2 + 1), {x, -2, 2}] ;

[Graphics:HTMLFiles/FamousCurves_162.gif]

Converted by Mathematica  (September 26, 2003)

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