The Bohemian Dome

Click for VRML

Introduction:

One could imagine creating a torus by taking a small ellipse and rotating it around a larger circle, smoothly turning the small ellipse to form a nice even tube. If, instead of turning the small ellipse as we rotate it, we force it to remain parallel to a certain plane (the yz plane in the plot above), the surface obtained is called a bohemian dome. The bohemian dome is one example of a type of surface known as a translation surface, the surface that results when one curve is rotated about a second curve in such a way that some point on the first curve traces the second curve.

The structure of the bohemian dome is easier to see when it is presented in a cutaway view:

Definition:

In this case, a is the radius of the larger circle, and b and c are the lengths of the axes of the ellipse.

Properties:

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

Mean curvature of the surface.
Surface colored by Mean curvature.