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Surface normal 208
If a (possibly non-flat) surface S is parameterized 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
If a surface S is given implicitly as the set of points satisfying , then, a normal at a point
on the surface is given by the gradient
since the gradient at any point is perpendicular to the level set, and (the surface) is a level set of
.
For a surface S given explicitly as a function of the independent variables (e.g.,
), its normal can be found in at least two equivalent ways. The first
one is obtaining its implicit form , from which the normal follows readily as the
gradient
.
(Notice that the implicit form could be defined alternatively as
;
these two forms correspond to the interpretation of the surface being oriented upwards or downwards, respectively,
as a consequence of the difference in the sign of the partial derivative .) The second way of obtaining the
normal follows directly from the gradient of the explicit form,
by inspection,
;
, where is the upward unit vector.
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.
Hypersurfaces in n-dimensional space
The definition of a normal to a surface in three-dimensional space can be extended to -dimensional
hypersurfaces in a -dimensional space. A hypersurface may be locally defined implicitly as the set of points
satisfying an equation , where is a given scalar function. If is continuously
differentiable, then the hypersurface obtained is a differentiable manifold, and its hypersurface normal can be
obtained from the gradient of , in the case it is not null, by the following formula