Cycloids - Two 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.
a	the radius of the circle, so a > 0.
w	increase w to show more revolutions in the same t-interval; w > 0.

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

a =1.25;
w =2.0;

x[t_]:=a*(w *t-Sin[w*t]);
y[t_]:=a*(1-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

xmin = a* (w *tmin - 1.0)
xmax=a *( w * tmax + 1.0)

dotsize = ((xmax - xmin)/(100.
))


ymin=0.0- dotsize
ymax=2*a + 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]},
{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)	 -a ≤ -a * Cos[w*t] ≤ a		Note that a>0.
(1c)	a - a ≤ a - a * Cos[w*t] ≤ a + a
(1d)	0 ≤ a - a * Cos[w*t] ≤ 2a
(1e)	0 ≤ y[t] ≤ 2a

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

(3a)	tmin ≤ t ≤ tmax
(3b)	 w * tmin ≤  w * t ≤  w * tmax 

(4a)	 w * tmin - 1 ≤   w * t - Sin[w*t] ≤ w * tmax + 1
(4b)	a * (w * tmin - 1) ≤  a * (w * t - Sin[w*t]) ≤ a * (w * tmax + 1)
(4c)	a *( w * tmin - 1) ≤  x[t] ≤ a *( w * tmax + 1)