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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