Intelligent Design Works presents

A Davis-Rutan Production

The Hypocycloid

A Sordid Tale of an Industrious Roulette

The Family

• Roulette- A roulette is the curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve.

– Trochoid- A subclass of roulettes, a trochoid is a case where both curves are circles. A point is fixed to the rotation of one circle which rolls along the other. • Hypotrochoid- A subclass of trochoids in

which the smaller circle is fixed to rotate about the interior of the larger.

The Family cont.

x ( )a b ( )cos t h

cos

( )a b t

b

y ( )a b ( )sin t h

sin

( )a b t

b

Parameterized with respect to t, the equations of the point P are given by the following equations:

The Hypotrochoids

HypocycloidHypocycloids are hypotrochoids in which h=b. Thus the point that traces the curve is attached to the rim of the smaller circle. Because of this equality, the loops of the previous hypotrochoid disappear.

An example of a Hypotrochoidwith three cusps:

A Hypocycloid of the sameorder as the previous example.

Properties of the Hypocycloid•The point P traces a number of cusps as the small circle rotates around the perimeter of the large circle. •The ratio of the radius of the large circle to the radius of the smaller circle will determine the number of cusps.

•For the above hypocycloid, the ratio of a to b is 4 to 1. Therefore, the curve has 4 cusps and is called an asteroid.

•The figure to the left represents the hypocycloid, where a is the radius of the outer circle and b is the radius of the inner. There exists such that

•As the inner circle rotates around the interior of the larger circle, the arcs S1 and S2 drawn by the points P and Q are equal.

The Derivation

ab

Hypocycloid cont.

•Therefore, the following is also true:

•Solving for β, we get

•If we consider the parametric equations of each circle, the motion of P can be represented by the position vector:

S2b b

a b b

S1 a

P [ ],( )a b ( )cos b ( )cos ( ) 1 ( )a b ( )sin b ( )sin ( ) 1

Examples of Hypocycloids

A hypocycloid where is25/9

A five pointed star generated by an of 5/3

Examples of Hypocycloids cont.

A hypocycloid with an of π (3.14……) Due to its irrational nature, the hypocycloids motion never repeats.

Sketchpad

We’ve used Geometer’s Sketchpad to make an animation of a hypocycloid. Follow the web address below to access and enjoy it.

http://online.redwoods.edu/instruct/darnold/calcproj/sp05/srutan/hypo2.gsp

Origins of the Hypocycloid

•The earliest known reference to hypocycloids appears in a 1525 textbook by mathematician and artist, Albrecht Dürer. The text was part I in a four part mathematics series titled Unterweisung der Messung mit dem Zirkel und Richtscheit. The book was the first mathematics text published in German.

•Also accredited with development of cycloids, Roemer and La Hire are said to have conceived applications of cycloids while engineering gear teeth in the 1600s.

Applications of the Hypocycloid

•In machines, it is often desirable to change rotational motion into translational motion. •In the 1800’s the hypocycloid came in handy for engineering train gears.

For a hypocycloid in which the inner circle has a radius half that of the outer circle, two cusps are created for translational motion.

References

[1] “MathWorld” http://mathworld.wolfram.com

[2] “Hypocycloid” http://www-groups.dcs.st-and.ac.uk/~history/curves/hypocycloid.html

[3] “Albrecht Durer” http://www-groups.dcs.st-and.ac.uk/~history/mathematicians/Durer.html

[4] “Kmodel” kmoddl.library.cornell.edumodel_metadata.php

Special Thanks To

~Dave Arnold~

for support and inspiration.(and for being awsome, of course)