Mathematical Curves

The Cartesian equation of Astroid is

x^2/3 + y^2/3 = a^2/3 or (x² + y² − 1)³ + 27x²y² = 0.

Parametric equations are

• x(t) = a cos³t = (3cos t + cos 3t)/4
• y(t) = a sin³t = (3sin t ⎼ sin 3t)/4.

Notable Facts:

• (a) If we consider a circle of radius a, this curve can be formed by rolling a circle of radius a/4 inside.
• (b) The curve can also be formed as the envelope produced when a line segment is moved with each end on one of a pair of perpendicular axes (glissette).
• (c) Also called the tetracuspid, since it has four cusps.