Intelligent Design Works presents

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Intelligent Design Works presents
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A Davis-Rutan Production
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The Hypocycloid
A Sordid Tale of an Industrious Roulette
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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.

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The Family cont.
The Hypotrochoids
Parameterized with respect to t, the equations
of the point P are given by the following
equations
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Hypocycloid
Hypocycloids are hypotrochoids in which hb. 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 Hypotrochoid with three cusps
A Hypocycloid of the same order as the previous
example.
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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.

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The Derivation

• 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.

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Hypocycloid cont.

• Therefore, the following is also true
• Solving for ß, we get
• If we consider the parametric equations of each
  circle, the motion of P can be represented by the
  position vector

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Examples of Hypocycloids
A hypocycloid where ? is 25/9
A five pointed star generated by an ? of 5/3
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Examples of Hypocycloids cont.
A hypocycloid with an ? of p (3.14) Due to its
irrational nature, the hypocycloids motion never
repeats.
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Sketchpad
Weve used Geometers Sketchpad to make an
animation of a hypocycloid. Follow the web
address below to access and enjoy it.
http//online.redwoods.edu/instruct/darnold/calcpr
oj/sp05/srutan/hypo2.gsp
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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.

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Applications of the Hypocycloid

• In machines, it is often desirable to change
  rotational motion into translational motion.
• In the 1800s 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.
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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/histo
ry/mathematicians/Durer.html 4 Kmodel
kmoddl.library.cornell.edumodel_metadata.php
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Special Thanks To

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