Math 105
Parametric Plots

These simulations show how circles on circles (big circles have little circles, on their paths to spiral, and little circles have littler circles, and so on ad infinitum) can lead to an incredible wealth ofshapes. Essentially we have a Taylor series of paths through circles!

?ParametricPlot

ParametricPlot[{Cos[t], Sin[t]},{t,0,2 Pi}]

Manipulate[ParametricPlot[{Cos[t], Sin[t]},{t,0,2 Pi c},PlotRange -> {-1.1,1.1}, Epilog -> {PointSize[0.04],Point[{Cos[2 Pi c], Sin[2 Pi c]}]}], {c,0.01,1}]

Manipulate[ParametricPlot3D[{Cos[t], Sin[t],t/20},{t,0,2 Pi c}], {c,1,10}]

Manipulate[ParametricPlot[{{Cos[t], Sin[t]},{Cos[t]+r Cos[t s], Sin[t] + r Sin[t s]}}, {t,0, 2 Pi c}, PlotRange -> {-2,2}, Epilog -> {PointSize[0.04],Point[{Cos[2 Pi c]+ r Cos[2 Pi c s], Sin[2 Pi c]+ r Sin[2 Pi c s]}]}], {c,.01,10}, {s,.1,10}, {r,.1,1}]

Manipulate[ParametricPlot[{{Cos[t], Sin[t]},{Cos[t]+r Cos[t s], Sin[t] + r Sin[t s]}}, {t,0, 2 Pi c}, PlotRange -> {-2,2}, Epilog -> {PointSize[0.04],Point[{Cos[2 Pi c]+ r Cos[2 Pi c s], Sin[2 Pi c]+ r Sin[2 Pi c s]}]}], {c,1,10}, {s,8,10}, {r,.2,1}]