The Sphere

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Introduction:

The sphere has the interesting property that is Gaussian and mean curvatures are constant. Because of its remarkable symmetry, though, this is hardly surprising. Note that, despite its relatively simple appearance, its parameterization in rectangular coordinates is somewhat complicated. An alternative to this parameterization is the use of spherical coordinates, in which the sphere is described more simply as shown in the second equation below.

Definition:

(spherical coordinates)

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.