Costa’s minimal surface is a thrice-punctured torus discovered by Celso José da Costa in 1982. It was the first new example of a complete, embedded minimal surface of finite topology found since the catenoid and helicoid in the 19th century.

Defining terms, a surface is minimal if it locally minimizes area, which is equivalent to having zero mean curvature everywhere. It is complete if every geodesic can be extended indefinitely within the surface (no edges or boundaries are “missing”). And it is embedded if it does not cross itself, as opposed to an immersed surface which may self-intersect. So basically, Costa’s surface is a smoothly self-avoiding, boundaryless minimal surface that lives in ordinary three-dimensional space.

The surface has two catenoidal ends (extending to infinity upward and downward) and a planar end (extending to infinity in the horizontal plane), giving it three “punctures” in its toroidal domain.

Hover the mouse or tap either diagram below to view the corresponding point on the other.

The surface is parameterized on the square torus (that is, the periodic unit square in the complex plane), with punctures at , , and — the three points where the surface extends to infinity, corresponding to the planar end and the two catenoidal ends respectively. For in the fundamental domain, the Weierstrass-Enneper representation gives:

where and are the Weierstrass elliptic and zeta functions for the square lattice, and is the value of at the midpoint of a real period. The Weierstrass functions are evaluated via Jacobi theta series in powers of the nome , which is small enough that the series converges in just a few terms.

The mesh is generated on the CPU by evaluating the parametrization on a grid and computing normals via finite differences. Triangles near the three punctures are excluded. Transparency is achieved through depth peeling.