Surface Normal Vector

Definition: SurfaceNormal Vector

Let $s:\mathcal{D}\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^3$ be a parametric surface which is differentiable at $\mathbf{a}$.

The normal vector of $s$ at $\mathbf{a}$ is the cross product of $s$‘s partial derivatives with respect to Cartesian coordinates:

$$\frac{\partial s}{\partial x}(\mathbf{a})\times \frac{\partial s}{\partial y}(\mathbf{a})$$

NOTATION

The normal vector is often denoted as $\mathbf{N}$ or $\mathbf{N}(\mathbf{a})$ and the unit normal vector as $\mathbf{n}$.

NOTE

The normal vector is sometimes also called just the surface normal.