The Gauss Map

Most of the extrinsic properties of a differentiable surface X(u,v) can be expressed in terms of the unit normal vector N(u,v) . Recall that a surface is called regular if the first partial derivative vectors Xu(u,v) and Xv(u,v) are linearly independent, so that the vector Xu(u,v) ·Xv( u,v) is non-zero. In this case we may define the unit normal vector, N(u,v), by the condition

As in the case of a curve, where we considered the velocity vectors to be based at the origin, we may consider the tangent space at the point X(u0,v 0) to be the plane through the origin defined by all linear combinations of the partial derivative vectors Xu(u,v) and Xv(u,v) . When these two vectors are linearly independent, they span a plane and the unit vector N(u,v) will be perpendicular to this plane.

The vector function N(u,v) is a map from R2 to R3, which can be considered a surface. Since

the surface N(u,v) maps the domain of X(u,v) to a portion of the unit sphere. The map N(u,v) is called the Gauss map of the surface, or the spherical image mapping.

Demonstration 1: Gauss Map of Selected Portions of Surfaces (square patch)
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Demonstration 1a: Gauss Map of Selected Portions of Surfaces (disc patch)
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In this demonstration, you can select a portion of the domain of a surface, and show the image on the surface in space and on the spherical image. The default surface is a torus, and you can select a point to be the center of a rectangular neighborhood with width and height determined by slider bars.