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Line or vector perpendicular to a curve or a surface
A polygon and its two normal vectors
A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.
For a surface in given as the graph of a function an upward-pointing normal can be found either from the parametrization giving
or more simply from its implicit form giving
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.
Choice of normal
A vector field of normals to a surface
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.
Note: in this section we only use the upper matrix, as translation is irrelevant to the calculation
When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.
Specifically, given a 3×3 transformation matrix we can determine the matrix that transforms a vector perpendicular to the tangent plane into a vector perpendicular to the transformed tangent plane by the following logic:
Write n′ as We must find
Choosing such that or will satisfy the above equation, giving a perpendicular to or an perpendicular to as required.
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, that is, purely rotational with no scaling or shearing.
where is a point on the hyperplane and for are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector in the null space of the matrix meaning That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation then the vector is a normal.
The normal line is the one-dimensional subspace with basis
Varieties defined by implicit equations in n-dimensional space
A differential variety defined by implicit equations in the -dimensional space is the set of the common zeros of a finite set of differentiable functions in variables
The Jacobian matrix of the variety is the matrix whose -th row is the gradient of By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank At such a point the normal vector space is the vector space generated by the values at of the gradient vectors of the
In other words, a variety is defined as the intersection of 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 of the variety is the affine subspace passing through and generated by the normal vector space at
These definitions may be extended verbatim to the points where the variety is not a manifold.
Let V be the variety defined in the 3-dimensional space by the equations
This variety is the union of the -axis and the -axis.
At a point where the rows of the Jacobian matrix are and Thus the normal affine space is the plane of equation Similarly, if the normal plane at is the plane of equation
At the point the rows of the Jacobian matrix are and Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the -axis.