Deltoid

A 3-cusped hypocycloid, also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner's hypocycloid (Lockwood 1967; Coxeter and Greitzer 1967, p. 44; MacTutor). The equation of the deltoid is obtained by setting in the equation of the hypocycloid, where is the radius of the large fixed circle and is the radius of the small rolling circle, yielding the parametric equations

			(1)
			(2)
			(3)
			(4)

The arc length, curvature, and tangential angle are

			(5)
			(6)
			(7)

The total arc length is computed from the general hypocycloid equation

(8)

With , this gives

(9)

The area is given by

(10)

with

(11)

The length of the tangent to the tricuspoid, measured between the two points , in which it cuts the curve again, is constant and equal to . If you draw tangents at and , they are at right angles.

Rather surprisingly, the deltoid can act as a rotor inside an astroid and, in fact, the deltoid catacaustic is an astroid.