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A surface normal, or simply '''normal''', to a normal to a Plane , the normal component of a Force , the '''normal vector''', etc. The concept of '''normality''' generalizes to Orthogonality .


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 Derivative s
:{\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




''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) .


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