The Superellipse: One Exponent From Astroid to Squircle to Rectangle

One exponent, many shapes

The ordinary ellipse equation (x/a)² + (y/b)² = 1 is a special case of a more general family discovered by Gabriel Lame in 1818: |x/a|ⁿ + |y/b|ⁿ = 1, where n can be any positive real number. Lame studied this purely as a mathematical curiosity; two centuries later the Danish scientist, inventor and poet Piet Hein revived it as a practical design tool, using it to shape a roundabout in Stockholm's Sergels Torg square in the 1960s where a plain ellipse looked too cramped and a rectangle too harsh.

What changes as n sweeps from 0 to infinity

n < 1   →  concave, star-like curve (astroid at n=2/3)
n = 1   →  diamond (rotated square, straight edges)
n = 2   →  ordinary ellipse (circle if a=b)
n = 4   →  Piet Hein's "squircle" - rounded-square superellipse
n → ∞  →  the bounding rectangle [-a,a]×[-b,b]

At n=2/3 with a=b the curve is exactly the astroid - the same four-cusped star that appears as the special-case hypocycloid with R=4r elsewhere on this site, confirming that these two independently-motivated curve families overlap at that one exponent. As n climbs past 2, the corners of the ellipse push outward and flatten, and the curve gradually approximates the bounding rectangle more and more closely without ever quite reaching it for any finite n - the limit is a genuine rectangle only in the n→∞ sense.

Piet Hein's squircle: why n=4

Hein specifically favoured n=4 (with a=b, giving a 'squircle') because it visually reads as rounder than a rounded rectangle but squarer than a circle - a genuinely intermediate shape rather than one that looks like a compromise. He applied it beyond the roundabout: to furniture, tableware and a puzzle called Superegg, a 3D solid of revolution based on the same exponent. The squircle later resurfaced in digital interface design, where app icons on some mobile platforms are drawn as superellipses rather than true rounded rectangles because the transition from flat edge to curve reads as smoother to the eye.

Parametrising the curve for rendering

The implicit equation is awkward to sample directly because solving for y given x involves a fractional power and a sign choice at every point. A parametric form avoids this, using signed powers of sine and cosine (denoted here with the notation cos^[n] meaning sign(cos t)·|cos t|ⁿ, preserving the sign that the naive power would destroy):

x(t) = a · sign(cos t) · |cos t|^(2/n)
y(t) = b · sign(sin t) · |sin t|^(2/n),   t ∈ [0, 2π)

Every point produced this way satisfies the implicit equation exactly and identically for any t, so a fixed number of samples (a few hundred) evenly spaced in t gives a smooth outline with no numerical root-finding required - a much cheaper approach than scanning x and solving for y at each column.

Comparing against a circle and a rounded rectangle

To make the shape's identity clear, the demo overlays the superellipse against two reference curves at the same bounding box: a true circle (equivalent to n=2) and a conventional rounded rectangle built from four quarter-circle arcs joining straight edges. The superellipse at n=4 visibly threads between them - its curvature changes continuously all the way around, unlike the rounded rectangle, whose curvature jumps abruptly from zero (along a straight edge) to a constant nonzero value (on an arc) at each of the eight tangent points. That continuous curvature is the mathematical reason the squircle looks more organic than a rounded rectangle despite occupying almost the same area.

Frequently asked questions

Try it live

Everything above runs in your browser — open Superellipse and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.