Hypocycloid

By

Nick Whitman


Description

     A hypocycloid is a special plane curve that is generated by the trace of a fixed point on a small circle that rolls within a larger circle.  It is very similar to the cycloid but instead of the circle rolling along a line it rolls within a circle.  The ratio of the radius of the larger circle to the radius of the smaller circle determines the number of cusps of the curve.  For example if the ratio is 3/1 the curve will hace three cusps and it will be a deltoid.

The circle with radius b rolls within the circle with radius a.  Point P is traced.

This is an illistration of the formation of a hypocycloid, namely the deltoid.

History

     The ordinary cycloid was discovered in 1599 by Galileo and Mersenne.  Hypocycloids were first concieved by Roemer in 1674 while he was studying the best form of gear teeth.  Johan Bernoulli worked with this curve in 1691.  Daniel Bernoulli discovered the double generation theorem of cycloidal curves in 1725.  Euler also did work with this curve in 1745, his work involved an optical problem.

These pictures show the formation of various hypocycloids.

 
 

Here are some hypocycloids with various ratios of radii.
The top four are those being constructed above.




Parameteric Equations

This is a picture of the small circle after it has rotated about pi/3 radians inside the larger circle.  The radius of the large circle is "a" and the radius of the small circle is "b".  The angle from the center of the large circle to the center of the small circle is "t".  The angle in radians that the small circle has rotated is "theta".

The parameteric equations for the center of the small circle as it rotates about the center of the large circle are:

     x=(a-b)*cos(t)     y=(a-b)*sin(t)

This is because it forms a circle with radius "(a-b)".

To get the parameteric equations of the hypocycloid you must add these parameteric equations to the parameteric equations of the point that is being traced.

This is a picture of the small circle only after it has rotated about pi/3 radians.  The  parameteric equations of point "P" (the point that is traced) on the circle measured with the center of this circle at the origin are:

     x=b*cos(theta)     y=-b*sin(theta)

Adding the two sets of parameteric equations you get the parameteric equations of the hypocycloid:

     x=(a-b)*cos(t)+b*cos(theta)     y=(a-b)*sin(t)-b*sin(theta)

Now we must replace "theta" with an expression in terms of "t".  This can be determined by looking at the difference in the periods of "t" and "theta".  We find that:

     theta=(a/b-1)*t

Replacing theta we have the parameteric equations of the hypocycloid:

     x=(a-b)*cos(t)+b*cos((a/b-1)*t)     y=(a-b)*sin(t)-b*sin((a/b-1)*t)

References

Weisstein, E. "Hypocycloid"   http://www.astro.virginia.edu/~eww6n/math/Hypocycloid.html

Lee, X. "Epicycloid and Hypocycloid"     http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html

Johnson, M. "The Deltiod Curve"   http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp98/MikeJ/deltiod.htm