Superellipse
The Lamé curve |x/a|ⁿ + |y/b|ⁿ = 1 that interpolates between ellipse and rectangle — the geometry of the squircle.
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Formulas and key values
- Cartesian equation
- Parametric equations
- AreaIn the graph ≈ 4.635
a and b are the semi-axes (b = 1 in the graph) and n is the exponent; n = 2 is an ellipse.
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Start the video01 · Definition
What is the Superellipse?
Gabriel Lamé generalized the ellipse in 1818 by freeing the exponent: |x/a|ⁿ + |y/b|ⁿ = 1. At n = 2 it is the ordinary ellipse; as n grows the sides flatten toward a rectangle; below n = 1 the sides cave inward, reaching a four-pointed star (the astroid, for a = b) at n = 2/3.
Danish designer Piet Hein made n = 2.5 famous in 1959 when he used it to shape Sergels Torg, a traffic roundabout in central Stockholm, arguing it blended the mechanical rectangle with the organic ellipse. The “squircle” near n = 4 now outlines tabletops, stadiums, and the rounded icons on your phone.
02 · Properties
Key properties
- n = 2: ellipse; n = 1: rhombus; n → ∞: rectangle; n = 2/3 with a = b: astroid.
- Parametrization: x = a·sgn(cos t)|cos t|^(2/n), y = b·sgn(sin t)|sin t|^(2/n).
- Enclosed area involves the Gamma function: 4ab·Γ(1+1/n)²/Γ(1+2/n).
- Always symmetric about both axes and convex for n ≥ 1.
- Piet Hein’s superegg — a revolved superellipse — balances stably on its end.
03 · Questions
Superellipse: common questions
- What is the equation of the Superellipse?
- Cartesian equation: |x/a|^n + |y/b|^n = 1; Parametric equations: x = a sgn(cos t) |cos t|^(2/n), y = b sgn(sin t) |sin t|^(2/n). a and b are the semi-axes (b = 1 in the graph) and n is the exponent; n = 2 is an ellipse.
- What is the area of the Superellipse?
- Area: A = 4ab (Γ(1 + 1/n)^2)/(Γ(1 + 2/n)). a and b are the semi-axes (b = 1 in the graph) and n is the exponent; n = 2 is an ellipse.
04 · Keep exploring