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

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

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

Trochoid

Trochoid is a family of curves.
(1) Let there be a point Q fixed to a circle. (it may be inside or outside of this circle)
(2) Let the circle roll on a line without slipping.
(3) The trace of Q is the trochoid.

If Q lies on the rolling circle, it is called a Cycloid. (see Cycloid).
If Q lies outside the rolling circle, it is sometimes called prolate or extended cycloid.
If Q lies inside the rolling circle, it is curate or contracted cycloid.

Let the radius of the rolling circle be r, and the distance from Q to the center of the circle be h.

Parametric formula: {r t - h Sin[t], r - h Cos[t]}. Periodicy: 2 Pi.

ParaPlot[ 
	{Trochoid[1, .4][t], Trochoid[1,1][t], Trochoid[1,2][t]},
	{t, -2 Pi, 4 2 Pi},
	PlotStyle->{Hue[0], Hue[.65], Hue[.4,1,.8]},
	AspectRatio->Automatic,
	PlotRange->{{-Pi, 5 Pi},{-2,4}}
];

ParametricPlot[
	Evaluate@ Table[ Trochoid[1,h][t], {h,0,(2 Pi-2), (2 Pi-2)/5}],
	{t, -3 Pi, 8 Pi},
	AspectRatio->Automatic,
	PlotStyle->Table[{Hue[0,i,1],Thickness[.007]},{i,.3,1,(.7)/6}],
	PlotRange->{{-Pi, 7 Pi}, {-3.8,5.8}},
	Background->Hue[.18]
];

?Troch*