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Return to Part V of the course APMA0330
Glossary
Famous Curves
Antiversiera
For two values of parameters:
a=-2; b=1;
and
a = 1; b = 2;
we have two graphs:


Arachnida
PolarPlot[ 2*a*Sin[n*\[Phi]]/Sin[(n - 1)*\[Phi]], {\[Phi], .0001, 2*\[Pi]}]

Astroid
ContourPlot[(x^2 + y^2 - t^2)^3 + 27*x^2*y^2 == 0, {x, -5, 5}, {y, -5, 5}]

Besace
ContourPlot[(x^2 - b*y)^2 + a^2*(y^2 - x^2) == 0, {x, -5, 5}, {y, -1.5, 4.5}, AspectRatio -> Automatic]

Bifolium
ContourPlot[(x^2 + y^2)^2 == b*x^2*y, {x, -1, 1}, {y, -0.2, 1}, AspectRatio -> Automatic]

Cardioid
PolarPlot[2*r*(1 - Cos[\[Phi]]), {\[Phi], 0, 2*\[Pi]}, AspectRatio -> Automatic]

Circular Tractrix
f[r_, th_] := th - ArcTan[Sqrt[4*a^2 - r^2]/r] - Sqrt[4*a^2 - r^2]/r
g[r_, th_] := {r Cos[th], r Sin[th]}
pl = ContourPlot[f[r, th] == 0, {r, 0, 8 Pi}, {th, 0, 4 Pi}, PlotPoints -> 30];
pl[[1, 1]] = g @@@ pl[[1, 1]];
Show[pl, PlotRange -> All, AspectRatio -> 1.5/2]

Cramer
ContourPlot[ x*(x^2 + y^2) == (r + l)*x^2 - (r - l)*y^2, {x, -1, 5}, {y, -5, 5}, AspectRatio -> 1]

Epicycloid
PolarPlot[Sqrt[ R^2 + h^2 - 2*(R + r)*h*Cos[R/r*\[Phi]]], {\[Phi], 0, 200*\[Pi]}]

Folium of Descartes
g[t_] = 3 t^2 /(1 + t^3);
ParametricPlot[{f[t], g[t]}, {t, 0, 20}, PlotRange ->All, AspectRatio -> 1, Plotlabel -> "Folium of Descartes", ImageSize ->200]

Galileo's Spiral
PolarPlot[a*\[Phi]^2 - l, {\[Phi], 0, 6*\[Pi]}]

Kiepert
![]() |
l = {1, 2, 3};
PolarPlot[(l^3*Cos[3*\[Phi]])^(1/3), {\[Phi], -2*\[Pi], 2*\[Pi]} |
Lemniscate
![]() |
F[t_] := 6*(Sec[t] Tan[t])/(1 +Tan[t]^3)
lemniscate = PolarPlot[F[t], {t,-Pi/6, 3*Pi/4.2}, PlotStyle -> {{Purple, Thickness[0.01]}}] ; shadingRight = ParametricPlot[{F[t]}, {t,0,CubeRoot[2]}, {r,0,F[t]}, PlotStyle -> {Red, Opacity[0.5]}, Mesh->None]; shadingLeft = ParametricPlot[{r*Cos[t], r*Sin[t]}, {t,0,CubeRoot[2]}, {r,0,F[t]}, PlotStyle -> {Green, Opacity[0.5]}, Mesh->None]; Show[lemniscate,shadingRight,shadingLeft] |
Limaçon

![]() |
a=3; l=3;
ContourPlot[(x^2 + y^2 - 2*a*x)^2 == l^2*(x^2 + y^2), {x, -1, 10}, {y, -7, 7}, AspectRatio -> 14/11] |
Rose
PolarPlot[a*Cos[k*\[Phi]], {\[Phi], 0, 4*\[Pi]}]


Trefoil
ParametricPlot[{r*(2*Cos[2*t] - Cos[t]), r*(2*Sin[2*t] + Sin[t])}, {t, 0, 10}]

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