• Introduction
• Case I: A and B are relatively prime.
  • Case II: A and B are not relatively prime.
• [1] Larson, Hostetler, and Edwards, Multivariable Calculus , 7th edition, Houghton Mifflin Company, 2002.

Hypocycloids and Epicycloids  Project

by Laurie L. Lacey, Ph.D., Schenectady County Community College, Schenectady NY, USA, laceyll@gw.sunysccc.edu, 2002 Laurie L. Lacey

This worksheet demonstrates the use of Maple for exploring plane curves and parametric equations beyond the specific examples found in a text.

Introduction

The chapter on parametric equations and plane curves found in the seventh edition of Calculus by Larson, Hostetler, and Edwards [1] contains an interesting section project on hypocycloids and epicycloids. Hypocycloids are defined by equations of the form     and         while epicycloids are defined by equations of the form        and           .  Maple allows the easy exploration of similar plane curves.  For example, in the following discourse the parametrized curves suggested in the above text are altered so that sin and cos are made to appear in the function defining x. The curves take on a mysterious appearance that has a certain aesthetic appeal.  In some cases the curves give the optical illusion of a hollow tube.  

The reader should note that,  xt(t), the function that yields the x coordinate of the plane curve, and yt(t), the function that yields the y coordinate of the plane curve, will be defined in general so that they can be used repeatedly. Two functions of A and B will be then defined from xt(t) and yt(t) that will allow the substitution of different numerical values of A and B.   Values of A and B were chosen arbritarily and the reader is encouraged to try other values than what appear here.  

>

restart;

Case I: A and B are relatively prime.

The first curves that were considered were generated by functions of the form

 and

There were only slight alterations made to the functions defining an epicycloid.  

Example 1:  In this example, A was chosen to have the value 8 while B was chosen to have the value 3.  These particular numbers were suggested by the project in the above mentioned text.

>	xt:=t->(A+B)*sin(t)-B*cos((A+B)/B*t);

>	yt:=t->(A+B)*sin(t)-B*sin((A+B)/B*t);

>	f:=subs(A=8,B=3,xt(t));

>	g:=subs(A=8,B=3,yt(t));

>	plot([f,g,t=0..14*Pi],scaling=constrained);

This was significantly different from the general appearance of the hypocycloid or of the epicycloid to pique curiosity.

Example 2:

In this example, A  and B were chosen relatively prime but the value of A was much greater than in the previous example while B remained 3.  

>	f:=subs(A=25,B=3,xt(t));

>	g:=subs(A=25,B=3,yt(t));

>	plot([f,g,t=0..14*Pi],scaling=constrained);

Observe that the resulting plane curve gives the appearance of a tube.

Example 3:

Experimentation with the values of A and B made some of the tubes look like Chinese finger puzzles.  

>	f:=subs(A=24,B=7,xt(t));

>	g:=subs(A=24,B=7,yt(t));

>	plot([f,g,t=0..14*Pi],scaling=constrained);

>

The reader is encouraged to have Maple solve for the values of t yielding horizontal tangents using  the command >solve(diff(g,t)=0,t);.  Some very interesting real numbers show up.

>

Case II: A and B are not relatively prime.

A and B are not relatively prime.

Example 4:  

>	f:=subs(A=20,B=12,xt(t));

>	g:=subs(A=20,B=12,yt(t));

>	plot([f,g,t=0..14*Pi],scaling=constrained);

This particular curve retained some appearance of a three dimensional object.  However, when B was selected to divide A the result was dramatically different.

Example 5:

>	f:=subs(A=12,B=3,xt(t));

>	g:=subs(A=12,B=3,yt(t));

>	plot([f,g,t=0..14*Pi],scaling=constrained);

Again, The reader is encouraged to have Maple solve for the values of t yielding horizontal tangents using  the command >solve(diff(g,t)=0,t);.

[1] Larson, Hostetler, and Edwards, Multivariable Calculus , 7th edition, Houghton Mifflin Company, 2002.

Disclaimer:  While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.