Cartesian equation:
(x2 + 2+12ax + 9a2)2 = 4a(2x + 3a)3
or parametrically:
x = a(2cos(t) + cos(2t)), y = a(2sin(t) - sin(2t))

Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

Click THIS LINK to experiment interactively with this curve and its associated curves.

The tricuspoid or 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.

The length of the tangent to the tricuspoid, measured between the two points P, Q in which it cuts the curve again is constant and equal to 4a. If you draw tangents at P and Q they are at right angles.

The length of the curve is 16a and the area it encloses is 2πa2.

In the parametric form the cusps occur at t = 0 , /3 and /3. Notice the similarity between the parametric form of the tricuspoid and the parametric form of the cardioid.

The pedal of the tricuspoid, where the pedal point is the cusp, is a simple folium. The pedal, where the pedal point is the vertex, is a double folium. If the pedal point is on the inscribed equilateral triangle then the pedal is a trifolium.

The caustic of the tricuspoid, where the rays are parallel and in any direction, is an astroid.

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Xah Lee

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JOC/EFR/BS January 1997

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