SURFACE NORMAL

A 'surface normal', or simply 'normal', to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point ''P'' on the surface is a vector perpendicular to the tangent plane to that surface at ''P''. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the 'normal vector', etc. The concept of 'normality' generalizes to orthogonality.

Contents

Calculating a surface normal

Uniqueness of the normal

Uses

''n''-dimensional surfaces

External link

Calculating a surface normal

For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.
For a plane given by the equation $ax+by+cz=d$, the vector $(a,\; b,\; c)$ is a normal. For a plane given by the equation 'r' = 'a' + α'b' + β'c', where 'a' is a vector to get onto the plane and 'b' and 'c' are non-parallel vectors lying on the plane, the normal to the plane defined is given by 'b' × 'c' (the cross product of the vectors lying on the plane).
If a (possibly non-flat) surface ''S'' is parametrized by a system of curvilinear coordinates 'x'(''s'', ''t''), with ''s'' and ''t'' real variables, then a normal is given by the cross product of the partial derivatives
:$\{partial\; mathbf\{x\}\; over\; partial\; s\}\; imes\; \{partial\; mathbf\{x\}\; over\; partial\; t\}.$
If a surface ''S'' is given implicitly, as the set of points $(x,\; y,\; z)$ satisfying $F(x,\; y,\; z)=0$, then, a normal at a point $(x,\; y,\; z)$ on the surface is given by the gradient
:$$
abla F(x, y, z).
If a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base. However, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

Uniqueness of the normal

A normal to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between the 'inward-pointing normal' and 'outer-pointing normal', which can help define the normal in a unique way. For an oriented surface, the surface normal is usually determined by the right-hand rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

Uses

★ Surface normals are essential in defining surface integrals of vector fields.

★ Surface normals are commonly used in 3D computer graphics for lighting calculations; see Lambert's cosine law.

''n''-dimensional surfaces

The definition of a normal to a two-dimensional surface in three-dimensional space can be extended to $n-1$-dimensional "surfaces" in $n$-dimensional space. Such a ''hypersurface'' may be defined implicitly as the set of points $(x\_1,\; x\_2,\; ldots,\; x\_n)$ satisfying the equation $F(x\_1,\; x\_2,\; ldots\; x\_n)\; =\; 0$. If $F$ is continuously differentiable, then the surface obtained is a differentiable manifold, and its surface normal is given by the gradient of $F$,
:$$
abla F(x_1, x_2, ldots, x_n) = left( frac{partial F}{partial x_1}, frac{partial F}{partial x_2}, ldots, frac{partial F}{partial x_n}
ight) .

External link

★ An explanation of normal vectors from Microsoft's MSDN