Gauss map

Figure 2.2: Surface and corresponding Gauss map along a curve.

Recall from sec:TangentPlane that given a parameterization $x$ of a regular surface $S$ , we can define a field of unique surface normals $N: S \rightarrow \ensuremath{\mathbb{R}^3}$

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

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 $p$ .