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A hypocycloid is a plane curve created by tracing a fixed point on a smaller circle as that circle rolls inside a larger circle without slipping.

Because this dance is performed by five dancers, they trace a five-cusped hypocycloid, sometimes referred to as a pentacuspid hypocycloid. Unlike the three-cusped deltoid and the four-cusped astroid, there is no widely used traditional special name for the five-cusped form.

A hypocycloid belongs to a wider family of mathematical curves known as roulettes, which are formed by one curve rolling along another. As the outer circle becomes larger relative to the inner one, the shape increasingly resembles a cycloid, the curve formed when a circle rolls along a straight line.

The shape of a hypocycloid depends on the ratio between the radii of the two circles. When this ratio is a whole number, the curve closes neatly and forms a set number of sharp points known as cusps. A three-cusped hypocycloid is called a deltoid, while a four-cusped version is known as an astroid. A special two-cusped form produces straight-line motion and is called the Tusi couple.

The Tusi couple was described in the 13th century by the Persian astronomer and mathematician Nasir al-Din al-Tusi. He used it in astronomical models to generate linear motion from rotating circles. Variations of the same principle later appeared in engineering and mechanical design.

Hypocycloids can be described mathematically using parametric equations. If the radius ratio is irrational, the curve never closes and instead continues to fill part of the region between the two circles. Mathematical studies of hypocycloids include their arc length, enclosed area, curvature, and related curves such as evolutes and involutes.

These curves also appear outside pure mathematics. The classic Spirograph drawing toy produces hypocycloid-related patterns, including hypotrochoids and epitrochoids. Certain logos and decorative designs use astroid forms because of their symmetry and distinctive geometry.

The Red Path Is A Four-Cusped 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).

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