Tricuspoid

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

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 , ^2π/₃ and ^4π/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.

Other Web site:

Xah Lee

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Tricuspoid.html