Surfaces of Revolution

A surface of revolution results from revolving a curve around a fixed axis. An example of a surface of revolution is the pseudosphere, which is the surface of revolution of a tractrix.

Minimal Surfaces

A minimal surface is one with zero mean curvature. Examples include Costa's minimal surface, Bour's surface, Catalan's minimal surface, the catenoid, Enneper's surface, the helicoid, Henneberg's surface, Richmond's surface, and Scherk's minimal surface.

Ruled Surfaces

If a surface has a 1-parameter family of straight lines, it is said to be ruled. The hyperboloid of one sheet can be parameterized as a ruled surface, as can Plücker's conoid.

Developable Surfaces

A ruled surface with no Gaussian curvature is said to be developable. The plane is an excellent example.

Tangent Surfaces

Tangent surfaces that are defined by

where alpha is a space curve, and v is the velocity vector for that curve.

Surfaces of Zero Curvature

It is possible for surfaces to have zero Gaussian curvature. The cylinder and the plane are prime examples.

Surfaces of Constant Curvature

Surfaces can have constant curvature despite their complicated appearences. Kuen's surface and Dini's surface, as well as the sphere and pseudosphere all have constant Gaussian curvature.

Conformal Surfaces

Two surfaces are said to be conformal if there exists an angle-preserving map between them.

Isometric Surfaces

Surfaces are called isometric when there exists a continuous deformation relating them. The helicoid and the catenoid are isometric; see the catenoid for an animation of the deformation.