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Surface normal 207
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that
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.
In the two-dimensional case, a normal line perpendicularly intersects
the tangent line to a curve at a given point.
The normal is often used in computer graphics to determine a surface's
orientation toward a light source for flat shading, or the orientation of
each of the corners (vertices) to mimic a curved surface with Phong
shading.
Calculating a surface normal
For a convex 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 , the
vector is a normal.
For a plane given by the equation
,
A polygon and two of its normal vectors
A normal to a surface at a point is the same as a
normal to the tangent plane to that surface at that
i.e., a is a point on the plane and b and c are (non-parallel) vectors lying on the plane, the normal to the plane is a
vector normal to both b and c which can be found as the cross product .
For a hyperplane in n+1 dimensions, given by the equation
,
where a 0 is a point on the hyperplane and a i for i = 1, ... , n are non-parallel vectors lying on the hyperplane, a normal
to the hyperplane is any vector in the null space of A where A is given by
.
That is, any vector orthogonal to all in-plane vectors is by definition a surface normal.
point.