Cycloids - Three parameters

Dr. Therese Shelton		shelton@southwestern.edu
Dr. Kendall Richards		richards@southwestern.edu

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.

Curves traced by a point rolling around a circle.

Suggestions to the instructor are given in this text style.  
Delete these portions to distribute as a student lab, 
which you may do acknowledging the source.

Caution:  remember that you may have to tweak some things if you change functions or graphing windows.

Items in yellow can be changed.

Set-up

Functions

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.

Clear[a,w,b, t, x,y]

a =1.25;
b =2.5;
w =2.0;

x[t_]:=a*w *t-b*Sin[w*t];
y[t_]:=a-b*Cos[w*t];

Define the graphing window.

Make everything decimal for the cleanest plotting. Allow room on the plot for the entire dot to show.

Clear[tmin,tmax,dotsize, xmin,xmax,ymin,ymax]

tmin=0.25
tmax=5 π/1.0

dotsize = ((tmax - tmin)/(75.
))

xmin = a* w *tmin - b- dotsize
xmax=a *w * tmax +b+ dotsize



ymin=a - b- dotsize
ymax=a +b + dotsize

Defining the plotting functions

Clear[p1]
p1[t_]:=ParametricPlot[{x[s],y[s]},{s,0,t},
Ticks->None,AspectRatio->Automatic,
PlotStyle->{Green},
DisplayFunction->Identity]

Clear[numberIntervals,deltat];
numberIntervals = 20.0;


deltat=((tmax - tmin)/numberIntervals)

Manipulate[
Show[p1[t],
Graphics[{
{Blue,Disk[{x[t],y[t]},dotsize]},
{Blue,Circle[{a*w*t,a},b]},
{Red,Circle[{a*w*t,a},a]}}],
Ticks->None,Axes->Automatic,AspectRatio->Automatic,Background->White,DisplayFunction->$DisplayFunction,
PlotRange->{{xmin,xmax},{ymin,ymax}}],{t,tmin,tmax,deltat}]

Double check the graph window settings and view the component functions.

Plot[x[t],{t,tmin,tmax},AxesLabel->{"t","x[t]"}]

Plot[y[t],{t,tmin,tmax},AxesLabel->{"t","y[t]"}]

You may wish to discuss how the plots above relate to the parametric plot.  These curves also help with discussions on the effects of changing the parameter values.

Inequalities for the graphing window

Some inequalities to help define the graphing window.

(1a)	  1 ≥  Cos[w*t]  ≥ -1
(1b)	 -b ≤ -b * Cos[w*t] ≤ b			Assume b > 0.
(1c)	a - b ≤ a - b * Cos[w*t] ≤ a+b
(1d)	a - b  ≤ y[t] ≤ a + b

(2a)	  1 ≥ Sin[w*t] ≥  -1
(2b)	 -b ≤ -b * Sin[w*t] ≤  b

(3a)	tmin ≤ t ≤ tmax
(3b)	a *w * tmin ≤  a * t ≤ a * w * tmax 		Assume a*w > 0. 

(4a)	a * w * tmin - b ≤  a * w * t - b * Sin[w*t] ≤ a * w * tmax + b
(4b)	a * w * tmin - b ≤  x[t] ≤ a * w * tmax + b