thumb|460px|The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving an [[astroid).]]

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. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line.

History

The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in Tahrir al-Majisti (Commentary on the Almagest). German painter and German Renaissance theorist Albrecht Dürer described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.

If <math>k</math> is a rational number, say <math>k = p/q</math> expressed as irreducible fraction, then the curve has <math>p</math> cusps.

To close the curve and complete the 1st repeating pattern:

  • <math>\theta</math>=0 to q rotations
  • <math>\alpha</math>=0 to p rotations
  • total rotations of rolling circle=p-q rotations

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

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

The area enclosed by a hypocycloid is given by:

<math display="block">A = \frac {(k - 1)(k - 2)} {k^2} \pi R^2 = (k - 1)(k - 2) \pi r^2 </math>

The arc length of a hypocycloid is given by:

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.

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.

alt=A circle with three hypocycloids inside|thumb|right|The Steelmark logo, featuring three hypocycloids

Curves similar to hypocycloids 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. Chilean soccer team CD Huachipato based their crest on the Steelers' logo, and as such features hypocycloids.

The first Drew Carey season of The Price Is Rights set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to high definition broadcasts starting in 2008, and only the giant price tag prop still features them today.

See also

  • Cyclogon
  • Epicycloid
  • Epitrochoid
  • Flag of Portland, Oregon, featuring a hypocycloid
  • Hypotrochoid
  • List of periodic functions
  • Murray's Hypocycloidal Engine, utilising a tusi couple as a substitute for a crank
  • Roulette (curve)
  • Special cases: Tusi couple, Astroid, Deltoid
  • Spirograph

References

Further reading

  • A free Javascript tool for generating Hypocyloid curves
  • Animation of Epicycloids, Pericycloids and Hypocycloids
  • Plot Hypcycloid &mdash; GeoFun
  • Iterative demonstration showing the brachistochrone property of Hypocycloid