Tangent plane and surface normal

In moving to the geometry of surfaces, many concepts are bootstrapped by examining the behaviour of curves within the surface, typically passing through some point of interest. For example, Appendix A uses curves to define the mapping of vectors from one space to another.

The tangent plane $T_p(S)$ at a point $p$ on a regular surface $S$ is defined as the subspace containing the tangents of all possible curves passing through $p$ . By the first two conditions of a regular surface (smoothness and non-self-intersection, see sec:RegularSurface), we are guaranteed that the curve tangents form a unique plane. Using the concept of a differential (Appendix A), the tangent plane can be defined as $dF_p(\ensuremath{\mathbb{R}^2})$ , the mapping of all possible tangent vectors at a point. Just as the differential does not depend on an underlying parameterization, neither does the definition of the tangent plane.

The surface normal is the unit vector perpendicular to all vectors in the tangent plane. There are two such vectors that satisfy the definition, and without a parameterization it is not possible to define a single normal vector, only a normal line. However, if a parameterization $x: (u,v) \rightarrow \ensuremath{\mathbb{R}^3}$ is given, then the partial derivatives $x_u$ and $x_v$ define a unique normal at a point $p$ via the rule

$$N(p) = \frac{x_u \times x_v}{\vert x_u \times x_v \vert}(p).$$ (2.11)

Note that the regularity condition guarantees that the denominator does not vanish. The field of normal vectors on the surface is can be shown to be continuous if the surface is orientable. Orientation of surfaces is not discussed further in these notes.