Hypocycloid

The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger blue circle (parameters are R=3.0, r=1.0, and so k=3, giving a deltoid).

In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

Properties

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:

$x (\theta) = (R - r) \cos \theta + r \cos \left( \frac{R - r}{r} \theta \right)$

$y (\theta) = (R - r) \sin \theta - r \sin \left( \frac{R - r}{r} \theta \right),$

or:

$x (\theta) = r (k - 1) \cos \theta + r \cos \left( (k - 1) \theta \right) \,$

$y (\theta) = r (k - 1) \sin \theta - r \sin \left( (k - 1) \theta \right). \,$

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable). Specially for k=2 the curve is a straight line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids, which had applications in the technology of high-speed printing press.

If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2r.

Each hypocycloid (for any value of r) is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius R.

Examples

• Hypocycloid Examples
• 

  k=3 - a deltoid

• 

  k=4 - an astroid

• 

  k=5

• 

  k=6

• 

  k=2.1

• 

  k=3.8

• 

  k=5.5

• 

  k=7.2

The hypocycloid is a special kind of hypotrochoid, which are a particular kind of roulette.

A hypocycloid with three cusps is known as a deltoid.

A hypocycloid curve with four cusps is known as an astroid.

Derived curves

The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself. [1]

The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve.

The isoptic of a hypocycloid is a hypocycloid.

Hypocycloids in popular culture

Curves similar to hypocyloids can be drawn with the Spirograph toy. Specifically, the Spirograph can draw hypotrochoids and epitrochoids.

The Pittsburgh Steelers' logo, which is based on the Steelmark, includes three astroids (hypocycloids of four cusps). In his weekly NFL.com column Tuesday Morning Quarterback, Gregg Easterbrook, often refers to the Steelers as the Hypocycloids.

The flag of Portland, Oregon features an astroid, a hypocycloid of four cusps.

The 2007 redesign of The Price is Right's set features astroids on the three main doors and the turntable area. [2]

See also

• Special cases: Astroid, Deltoid
• Cycloid
• Epicycloid
• Hypotrochoid
• Epitrochoid
• Spirograph

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 168,171–173. ISBN 0-486-60288-5. 

External links

• City of Portland Flag
• A free Javascript tool for generating Hypocyloid curves