Lamé curve

A Lamé curve is any of a family of curves related to the ellipse and that was first recognized and studied in 1818 by the French physicist and mathematician Gabriel Lamé (1795–1870). The formula for the Lamé curve family is a generalization of the equation for an ellipse (|x / a|2 + |y / b|2 = 1), namely:


    |x / a|n + |y / b|n = 1,


where n is any real number.


When n = 0, the curve reduces to a pair of crossed lines. As n increases, the curve changes from a curved star shape to a rectangle, with diagonals 2a and 2b, when n = 1. The special case when n = 2/3 corresponds to the astroid. Between n = 1 and n = 2 the curve turns from a curved rectangle to an ellipse (or a circle when both a and b are 1). For values of n greater than 2, Lamé curves are known as superellipses.