Hypocycloid

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Parametric Cartesian equation: x = (a - b)cos(t) + bcos((a/b - 1)t), y = (a - b)sin(t) - bsin((a/b - 1)t)

This is a member of a set of four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point P on a circle of radius b which rolls round a fixed circle of radius a.

For the hypocycloid, shown above in dodgy ASCII, the circle of radius b rolls on the inside of the circle of radius a. The point P is on the circumference of the circle of radius b. For the example a = 5 and b = 3.

Special cases

• a = 3b, a tricuspoid is obtained.
• a = 4b, an astroid is obtained.
• a = (n + 1)b, where n is an integer, the length of the epicycloid is 8nb and its area is b²(n² - n)
• The pedal curve, when the pedal point is the centre, is a rhodonea curve.

Also, the evolute of a hypocycloid is a similar hypocycloid.

Hy`po*cy"cloid (?), n. [Pref. hypo- + cycloid: cf. F. hypocycloide.] Geom.

A curve traced by a point in the circumference of a circle which rolls on the concave side in the fixed circle. Cf. Epicycloid, and Trochoid.

© Webster 1913.