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flat Surface is a three-dimensional Vector which is Perpendicular to that surface. A normal to a non-flat surface at a point ''p'' on the surface is a vector which is perpendicular to the Tangent Plane to that surface at ''p''. The word ''normal'' is also used as an adjective as well as a noun with this meaning: a line ''normal'' to a plane, the ''normal'' component of a force, the ''normal vector'', etc. ''A polygon and its normal'' CALCULATING A SURFACE NORMAL For a Polygon (such as a Triangle ), a surface normal can be calculated as the vector Cross Product of two edges of the polygon. For a Plane given by the equation , the vector is a normal. 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 : 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 : 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 an Oriented Surface , the surface normal is usually determined by the Right-hand Rule . USES
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