Lamé Curves / Superellipses

Gabriel Lamé's family: |x/a|^n + |y/b|^n = 1. The exponent n continuously morphs the shape from a four-pointed star (n<1) through a diamond (n=1), ellipse (n=2), to rounded square (n→∞). Piet Hein used superellipses in urban design; n=2.5 became the Stockholm roundabout.

n = 2.00
Shape: Ellipse
Area: π·a·b
Perimeter:
|x/a|^n + |y/b|^n = 1

n<1 → astroid family
n=1 → rhombus / diamond
n=2 → ellipse / circle
n→∞ → rectangle