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.

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

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

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

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

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

$$= <x_n,y_n,z_n> \cdot < x_q - x_p, y_q - y_p, z_q - z_p >$$

$$= x_n ( x_q - x_p ) + y_n ( y_q - y_p ) + z_n ( z_q - z_p )$$

and so we can write,

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

which is in the form

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

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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