The normal is often used in 3D computer graphics notice the singular, as only one normal will be defined to determine a surface's orientation toward a light source for flat shading , or the orientation of each of the surface's corners vertices to mimic a curved surface with Phong shading. 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.
Since a surface does not have a tangent plane at a singular point , it has no well-defined normal at that point: for example, the vertex of a cone.
In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous. The normal to a hyper surface is usually scaled to have unit length , but it does not have a unique direction, since its opposite is also a unit 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.
For an oriented surface , the normal is usually determined by the right-hand rule or its analog in higher dimensions. If the normal is constructed as the cross product of tangent vectors as described in the text above , it is a pseudovector.
Urban Dictionary: normal
When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals. Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, i. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. The definition of a normal to a surface in three-dimensional space can be extended to n -1 -dimensional hypersurfaces in R n.
At these points a normal vector is given by the gradient:. A differential variety defined by implicit equations in the n -dimensional space R n is the set of the common zeros of a finite set of differentiable functions in n variables. By the implicit function theorem , the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank k.
At such a point P , the normal vector space is the vector space generated by the values at P of the gradient vectors of the f i. In other words, a variety is defined as the intersection of k hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. The normal affine space at a point P of the variety is the affine subspace passing through P and generated by the normal vector space at P.
These definitions may be extended verbatim to the points where the variety is not a manifold. At the point 0, 0, 0 the rows of the Jacobian matrix are 0, 0, 1 and 0, 0, 0.
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Being made acutely aware of the privilege her life afforded her even without the wealth and celebrity she accumulated was another. In what ways do I support and uphold a system that is structurally racist? How do my race, class and gender affect my perspective? But instead of seeing these differences as divisive, I could have asked whether defining them was actually empowering and bringing about better understanding.
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