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.

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

 0

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