# deltoid

A deltoid is a hypocycloid with three cusps,
also known as a **tricuspoid** or **Steiner's hypocycloid** after the Swiss mathematician Jakob Steiner who investigated the curve in 1856.

The deltoid, so-named because it looks like an uppercase Greek delta (Δ), is formed by a point on the circumference of a circle rolling inside another circle with a radius three times as large. Among the first to study its properties was Leonhard Euler while working on a problem in optics in 1745.

The parametric equations of the cycloid with inner circle of radius *r* are:

*x*(*t* ) = 2*r* cos *t* + *r* cos
2*t*

*y*(*t* ) = 2*r* sin *t* - *r* sin 2*t*

The length of the path of the deltoid is 16*r*/3, and the area inside
the deltoid is 2π*r*^{ 2}. 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* equals 4*r*. If the deltoid's tangents
are drawn at points *A* and *B*, they will be perpendicular, and
they will intersect at a point inside the deltoid that is the 180° rotation
of point *P* about the center of the fixed circle.