Superellipse
Parametric curvesThe Lamé curve |x/a|ⁿ + |y/b|ⁿ = 1 that interpolates between ellipse and rectangle — the geometry of the squircle.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
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.
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.
Where you'll see it
App icon outlines, Sergels Torg in Stockholm, stadium and table design, and font glyph engineering.