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 (commonly called the normal vector) and that pass through a point . 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.

For a postscript 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 and that pass through a point . The point and the vector uniquely define the plane. Let be the plane defined by and . Then for any point on the plane, we must have that

since the vector will be in the plane. This relationship is illustrated in the following figure.

A Plane Equation

Suppose we are given a plane defined by a point and a vector . If we write the vector as , the point as , and an arbitrary point on the plane as , then from the above we have that

and so we can write,

which is in the form

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.

This document maintained by Ken Joy Comments to the Author All contents copyright (c) 1996, 1997 Computer Science Department, University of California, Davis All rights reserved.