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On-Line Computer Graphics Notes
EQUATIONS OF A PLANE


Overview

A plane in three-dimensional space is the locus of points that are perpendicular to a vector $ {\vec n} $ (commonly called the normal vector) and that pass through a point $ {\bf P} $. They form the fundamental geometric structure for many operations in computer graphics ( e.g., clipping) and geometric modeling ( e.g., tangent planes to surfaces). Two equivalent definitions of a plane are used and we present both in these notes.

pdficonsmall.gif For a pdf version of these notes look here.


Specifying a Point and a Vector

A plane in three-dimensional space is the locus of points that are perpendicular to a vector $ {\vec n} $ and that pass through a point $ {\bf P} $. The point and the vector uniquely define the plane. Let $ {\cal P} $ be the plane defined by $ {\vec n} $ and $ {\bf P} $. Then for any point $ {\bf Q} $ on the plane, we must have that

$\displaystyle {\vec n} \cdot < {\bf Q} - {\bf P} > \: = \: 0
$

since the vector $ < {\bf Q} - {\bf P} >$ will be in the plane. This relationship is illustrated in the following figure.

\includegraphics {figures/definition-of-a-plane}


A Plane Equation

Suppose we are given a plane defined by a point $ {\bf P} $ and a vector $ {\vec n} $. If we write the vector $ {\vec n} $ as $ {\vec n} = <x_n,y_n,z_n>$, the point $ {\bf P} $ as $ {\bf P} = (x_p,y_p,z_p)$, and an arbitrary point $ {\bf Q} $ on the plane as $ {\bf Q} = (x_q,y_q,z_q)$, then from the above we have that

0 $\displaystyle = {\vec n} \cdot < {\bf Q} - {\bf P} >$    
  $\displaystyle = <x_n,y_n,z_n> \cdot < x_q - x_p, y_q - y_p, z_q - z_p >$    
  $\displaystyle = x_n ( x_q - x_p ) + y_n ( y_q - y_p ) + z_n ( z_q - z_p )$    

and so we can write,

$\displaystyle x_n x_q + y_n y_q + z_n z_q - ( x_n x_p + y_n y_p + z_n z_p ) \: = \: 0
$

which is in the form

$\displaystyle Ax+By+Cz+D = 0
$

which is a common expression of the equation of a plane. We will both forms of this definition in the clipping algorithms of the viewing pipeline.


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Computer Science Department
University of California, Davis

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Ken Joy
1999-12-06