3D graphics eBook - Course Materials Repository

Surface normal 209
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
A vector field of normals to a surface
the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.
Transforming normals
When applying a transform to a surface it is sometimes convenient to deriving normals for the resulting surface from
the original normals. All points P on tangent plane are transformed to P′. We want to find n′ perpendicular to P. Let
t be a vector on the tangent plane and M l be the upper 3x3 matrix (translation part of transformation does not apply
to normal or tangent vectors).
So use the inverse transpose of the linear transformation (the upper 3x3 matrix) when transforming surface normals.
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.
• Surface normals are often adjusted in 3D computer graphics by normal mapping.
• Render layers containing surface normal information may be used in Digital compositing to change the apparent
lighting of rendered elements.