deltoid

This curve, with the form of the Greek capital delta, has been given also as names:

• Steiner's hypocycloid: the Swiss mathematician Jakob Steiner studied the curve in 1856
• tricuspoid: because of its three cusps

This quartic curve ¹⁾  is the hypocycloid for which the rolled circle is three times as large as the rolling circle.
Given a tangent l to the curve. It cuts the curve in points P and Q. Then PQ has a length of 4, and the tangents to the curve in P and in Q make a right angle.
The length of the deltoid is 16, and the area it encloses: 2π.

Some other properties of the curve:

• the radial of the deltoid is the trifolium
• its catacaustic (with the source at infinity) is the astroid
• its orthoptic is the circle
• the deltoid is the envelope of Simson lines

Three pedals of the curve are:

The curve has been investigated as first by Leonhard Euler (1745), while studying an optical problem.

notes

1) In Cartesian coordinates: (x² + y²)² -8x(x² - 3y²) + 18(x² + y²) = 27