The Torus

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Introduction:

The torus shown above is known as a ring torus, because its wheel radius a exceeds its tube radius b. When a and b are equal, the torus is called a spindle torus, and when b > a, it is referred to as either an inside-out torus or a horn torus.

Definition:

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:

Note that the curvature depends only on whether you are "inside" or "outside" the torus, not on your position around the rim.

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

Mean Curvature:

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