L-36.com

Equation of a plane



Written by Paul Bourke
March 1989


The standard equation of a plane in 3 space is

Ax + By + Cz + D = 0

The normal to the plane is the vector (A,B,C).

Given three points in space (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) the equation of the plane through these points is given by the following determinants.

Expanding the above gives
A = y1 (z2 - z3) + y2 (z3 - z1) + y3 (z1 - z2)
B = z1 (x2 - x3) + z2 (x3 - x1) + z3 (x1 - x2)
C = x1 (y2 - y3) + x2 (y3 - y1) + x3 (y1 - y2)
- D = x1 (y2 z3 - y3 z2) + x2 (y3 z1 - y1 z3) + x3 (y1 z2 - y2 z1)

Note that if the points are colinear then the normal (A,B,C) as calculated above will be (0,0,0).

The sign of s = Ax + By + Cz + D determines which side the point (x,y,z) lies with respect to the plane. If s > 0 then the point lies on the same side as the normal (A,B,C). If s < 0 then it lies on the opposite side, if s = 0 then the point (x,y,z) lies on the plane.



NOTICE: Some pages have affiliate links to Amazon. As an Amazon Associate, I earn from qualifying purchases. Please read website Cookie, Privacy, and Disclamers by clicking HERE. To contact me click HERE. For my YouTube page click HERE