Pseudospherical Surfaces Gallery

Surfaces in this gallery have a constant, negative Gauss curvature (K). The most well-known example of such a surface is the pseudosphere. which is obtained by rotating the plane curve known as a tractrix around its asymptote. (To see a pseudosphere, select the "Dini Surface Family" exhibit, and set the value of the parameter aa to 0.5 using the "Set Parameters" command in the "Settings" menu.)

To define the curvature of a surface at a point on the surface, consider a plane that is perpendicular to the surface and passes through that point. The intersection of that plane with the surface is a curve that contains the surface point. Consider the curvature of the curve at that point. (This curvature can be defined, for example, as 1/r, where r is the radius of the circle that best approximates the curve at that point, the so-called osculating circle.) Different planes will produce different curvatures, but among all the curvatures, there are a maximum value and a minimum value. The minimum and maximum values are called the principal curvatures of the surface at the given point, and the Gaussian curvature at that point is defined as the product of the two principal curvatures.

Several of the exhibits in this gallery are defined as parametric surfaces. The others -- those below the separator in the menu -- are defined by partial differential equations and can take quite a while to compute.

The Surfaces of Curvature K=1 gallery is one of several sub-galleries of the Surfaces Gallery. For more information on parametric surfaces, you should see the documentation for the Surfaces Gallery.