WOLFRAM|DEMONSTRATIONS PROJECT

Tensor Equation of a Plane

​
q
'
i
1
1
2
0
3
0
q
''
j
1
0
2
1
3
0
q
'''
k
1
0
2
0
3
1
O
l
1
-1
2
1
3
1
x+y+z
=
1
λ(O, P)
=
0
The general equation of a plane can be written as
A
q
1
+B
q
2
+C
q
3
+D=0
, where coefficients
A
,
B
,
C
, and
D
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
A
,
B
,
C
, and
D
. These equations can be rewritten together using the antisymmetric permutation tensor
ijkl
ϵ
; however, the three-vectors
q
1
,
q
2
,
q
3
for points in
3

and the origin
O
need to be rewritten as four-vectors so that they have a compatible dimension. The first three components of the vectors
q'
i
,
q''
j
,
q'''
k
, and
O
l
are equal to the coordinates of a point in
3

and the fourth component of these vectors is
1
. Similarly, the first three components of the vector
e
are equal to the free variables
q
1
,
q
2
, and
q
3
, while the fourth component of this vector is
1
. The matrix
η
lq
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
ijkl
ϵ
q'
i
q''
j
q'''
k
e
l
=0
.
A point-to-plane distance function can also be written in tensor notation. The distance
λ
between an origin
O
and a plane
P
can be written as:
λ(O,P)=
ijkl
ϵ
q'
i
q''
j
q'''
k
O
l

-1

ijkl
ϵ
q'
i
q''
j
q'''
k
η
lq
mnpq
ϵ
q'
m
q''
n
q'''
p

.
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.