A superellipse (or Lamé curve) is the set of points satisfying |x/a|^n + |y/b|^n = 1, where a and b are the half-axes and the exponent n > 0 controls the shape.

Parametric form

To draw it smoothly we use the angle parametrisation:
x = a·sgn(cosθ)·|cosθ|^(2/n)
y = b·sgn(sinθ)·|sinθ|^(2/n)

Sweeping θ from 0 to 2π traces the full closed curve for any n.

Morphing with n

• n < 1 — concave (pinched) sides.
• n = 2/3 — the astroid, x²ᐟ³+y²ᐟ³=a²ᐟ³.
• n = 1 — a diamond (rhombus / taxicab circle).
• n = 2 — an ordinary ellipse (a circle when a = b).
• n ≈ 2.5 — the squircle: rounded yet full-bodied.
• n → ∞ — approaches a rectangle (Chebyshev / sup norm).

Piet Hein & the squircle

In 1959 the Danish poet–scientist Piet Hein used the superellipse with n = 2.5 to lay out the roundabout at Sergels Torg in Stockholm — a shape that is neither a harsh rectangle nor a wasteful oval. The same squircle later shaped his furniture and, decades on, the rounded icons of modern phone interfaces such as iOS app tiles.

Superellipsoid

Extending the idea to three dimensions gives the superellipsoid, widely used in computer graphics and robotics to model rounded box-like solids with a single shape parameter.