The Superellipse: A Generalization of Ellipses and Rectangles

What is a Superellipse?

A superellipse is a geometric figure defined by the equation φ(x,y) = |x/a|^n + |y/b|^n = 1, where n and (a,b) are parameters. When n=2, it reduces to an ellipse; when n=1, it becomes a rectangle. The parameter n controls how circular or angular the shape is.

Superellipses were introduced by Hans Hahn in 1954 but gained popularity through Piet Hein's use of them in architecture and design.

Properties of Superellipses

The area A and perimeter P of a superellipse can be calculated using complex integrals, which depend on the parameters n, a, and b. For example, the area is given by A = 4ab π^(1/n) / α(n), where α(n) is the complete elliptic integral of the first kind.

By adjusting the exponent n, one can smoothly transition between a circle (when n approaches infinity) and a square (when n=1). This versatility makes superellipses useful in various fields including architecture, typography, and even video game design.

Real-World Applications

Superellipses have found applications in urban planning where they can be used to create more flexible and aesthetically pleasing shapes for public spaces. In typography, superellipses are used to design typefaces that blend the sharp angles of square letters with the smooth curves of circular ones.

In video game design, superellipses allow for the creation of non-traditional level layouts that can enhance gameplay experience by providing unique and challenging environments.

Interactive Exploration

By adjusting the exponent n in a superellipse, one can observe how the shape transitions from a circle to a square. The aspect ratio (a/b) controls the elongation of the shape along each axis.

The interactive nature of exploring superellipses helps in understanding not only their mathematical properties but also their practical applications in design and architecture.

Frequently asked questions

Try it live

Everything above runs in your browser — open Superellipse: 2D and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.