6.1: The Gauss MapMost 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) x 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,v0) 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
Demonstration 1: Gauss Map of Selected Portions of Surfaces (square patch) 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. The size of the square patch can be changed by clicking and dragging the white dot located at one of the square's corners.
Demonstration 1a: Gauss Map of Selected Portions of Surfaces (disc patch) This demonstration also uses a torus as its default surface but allows you to look at a circular neighborhood of the surface. This circular patch can be moved around and resized using the two white dots that appear in the domain window. Together the square patch and circular patch demos can give a good description of how the normal map takes a portion of a surface onto the unit sphere. Exercises
1. What does the Gauss mapping do to a portion of the outside part of the torus? |