Cycloids - Three parameters

Dr. Therese Shelton	
Southwestern University
Georgetown, TX

From Susan Colley's Vector Calculus, 3rd Edition, 2006, Pearson Education, Inc., Pearson Prentice Hall, ch 1:  

A fixed point of interest lies on the edge of a circle.  For instance, consider a reflector  is positioned at the edge of a bicycle wheel.  The wheel rolls along a straight line in a plane.  As the red wheel rolls along, the reflector is a blue dot that traces a green curve.

Define the parameters and functions for the cycloid.  All are positive.
a	the radius of the circle rolling along the line
w	increase w to show more revolutions in the same t-interval
b	the distance from the circle's center to the point of interest; i.e., the radius of the circle with a reflector on it.

b < a:  the point is inside the circle.
Consider a reflector positioned on a spoke within a bicycle wheel.  The wheel rolls along a straight line in a plane.  As the wheel rolls along, the reflector is a dot that traces a curve.
The point traces a curate cycloid. 

b = a:  the point is on the edge of the circle.  
Consider a wad of gum positioned at the edge of a bicycle wheel.  The wheel rolls along a straight line in a plane.  As the wheel rolls along, the gum is a dot that traces a curve.
The point traces a cycloid.

b > a:  the point lies on another circle of larger radius.  
Imagine two wheels,both centered on the same axle,one of larger radius than the other.  A straight, flat track is attached to a wall at a level higher than the larger radius.The smaller wheel runs on the track,and a reflector is positioned at the edge of a the larger wheel.As the wheels roll along,the reflector is a dot that traces a curve.
The point traces a prolate cycloid or epicycloid.

Manipulate[ParametricPlot[{a*w *t-b*Sin[w*t],a-b*Cos[w*t]},{t,0,16},
Ticks->None,AspectRatio->Automatic,PlotStyle->{Green},
Axes->True,Background->White,PlotRange->{{-3,30},{-3,5}}],
{a, 0.5,2.0,.5},{b, 0.5,2.0,.5},{w, 1,3,1}]