Gauss map

Figure 2.2: Surface and corresponding Gauss map along a curve.
Image GaussMap
Recall from sec:TangentPlane that given a parameterization $ x$ of a regular surface $ S$ , we can define a field of unique surface normals $ N: S \rightarrow \ensuremath{\mathbb{R}^3}$

$\displaystyle N(p) = \frac{x_u \times x_v}{\vert x_u \times x_v\vert}.
$

The space of all possible normals lies in the unit sphere in $ \mathbb{R}^3$ , so we can identify with each normal its two-dimensional location in the sphere. This map $ N: S \rightarrow \ensuremath{\mathbb{R}^2}$ is called the Gauss map. The Gauss map is differentiable and the differential $ dN_p$ maps vectors in the tangent plane to vectors in the tangent plane. That is, given a point $ p$ on the surface and a direction $ v$ in its tangent plane, $ dN_p(v)$ gives the change in surface normal as you move from $ p$ to $ p + \epsilon v$ . The change of the surface normal is a vector again in the tangent plane of $ S$ at $ p$ . Hence the differential $ dN_p$ gives the change in surface normal in the neighbourhood of $ p$ .

Copyright © 2005 Adrian Secord. All rights reserved.