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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+