## 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.