Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]
(*MathSource Item:0207-986*)

Needs["PlaneCurveFormulas`"]


Needs["PlaneCurveGenerator`"]

Needs["TrochoidPlot`"]
(*MathSource Item:0207-234*)

Needs["TrochoidShow`"]
(*MathSource Item:0207-997*)

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

"spiral graphs" movie

e

An animation template of a hypotrochoid with parameter h varing. You can try varing other parameters. You may want to set a fixed PlotRange to see the true scaling of the curve.

(* "Evolving Spiralgraph" *)
(* typical hypotrochoid movie with parameter h varing. *)
Do[
	ParametricPlot[
		Evaluate[{
			Hypotrochoid[1,6/13,i][t]
		}],
		{t,0, 6 2 Pi}, Axes->False, PlotRange->All,
		AspectRatio->Automatic, PlotStyle->{Hue[Random[],1,.5]}
	],
	{i, 6/13/3, 1+1/3, (1+1/3-6/13/3)/20}
]

e

Here is a fun animation based on two rotating hypotrochoids, creating a "flashing" illusion.

(* "Flashing illusion", two rotating hypotrochoids *)
Do[
	ParametricPlot[
		Evaluate[{
			Rotate[Hypotrochoid[1,1/7,2][t], i],
			Rotate[Hypotrochoid[1,1/7,2][t], -i]
		}],
		{t,0, 2 Pi},
		PlotStyle->{Hue[(i-(1/7 2 Pi/15))/((1/7 2 Pi)-(1/7 2 Pi/15)),1,.9]},
		AspectRatio->Automatic, Axes->False,
		PlotRange->{{-1,1},{-1,1}} 3, PlotStyle->{Hue[0]}
	], {i, 1/7 2 Pi/15, 1/7 2 Pi, 1/7 2 Pi/20}
]