Deltoid
(hypocycloid of three cusps)

Thedeltoid, or tricuspoid, is a shape formed by a point on a circle rolling inside another circle with a radius three times as large. Thedeltoid will  also be the path if the inner circle is 2/3 the radiusof the outer.

The deltoid is also related to the Simson Line.  If the Simson Line is drawn for all points around the circumcircle of a triangle, the envelope of the lines is a deltoid.

The parametric equations of the cycloid with innner circle of radius r, and outer circle of radius 3r  are given by x(t)=2rcos(t)+r cos(2t), and y(t)=2r sin(t)-r sin(2t). The image at right shows the graph created by a Ti-83 calculator with r=1 and the fixed circle of radius three around the deltoid.

If a tangent is drawn to the deltoid at some point, P, and the points where the tangent crosses the deltoids other two branches are called points A and B, then the length of AB is equal to 4 times r.

If you draw the deltoids tangents at points A and B, they will be perpendicular, and they will intersect at a point inside the deltoid that is the 180 degree rotation of point P about the center of the fixed circle. The length of the path of the deltoid is 16r/3, and the area inside the deltoid is 2pr2.
The name for the deltoid comes from the fact that it looks like delta, D , the fourth letter of the Greek alphabet, which looks like an isosceles triangle. Prior to the moder use of the word kite, deltoid was used as a name for that figure because it looked like two isosceles triangles put together with a common base. The word also persists today in more common usage as the name for the triangular shaped muscle covering the shoulder joint. Euler was one of the first to study the properties of the curve in his work on optics.