Tensor Equation of a Plane

The general equation of a plane can be written as , where coefficients , , , and are determined using a set of matrices.
The simplification of the plane equations to one tensor equation proceeds from the similarity of the three-vector equations for , , , and . These equations can be rewritten together using the antisymmetric permutation tensor ; however, the three-vectors , , for points in and the origin need to be rewritten as four-vectors so that they have a compatible dimension. The first three components of the vectors , , , and are equal to the coordinates of a point in and the fourth component of these vectors is . Similarly, the first three components of the vector are equal to the free variables , , and , while the fourth component of this vector is . The matrix allows the first, second, and third components of a four-vector to be selected for summation.
Choosing three distinguishable points, the equation for a plane can be written in tensor notation as .
A point-to-plane distance function can also be written in tensor notation. The distance between an origin and a plane can be written as:
.
Tensor notation for planes and distances could be very useful to material scientists who would like to define coordinate systems and compute distance functions without using Miller indices.

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