Graphics Reference
In-Depth Information
4 The Plane
4.1 Introduction
Planar polygons play an important role in computer graphics. In this chapter we discover
what benefits vectors bring to solving problems involving planes. We begin this exploration by
examining the Cartesian form of the plane equation.
4.2 The Cartesian form of the plane equation
The general form of the plane equation creates an equation that equals zero, whereas the
Cartesian form organizes the equation such that a constant term is isolated on one side of the
equals sign:
+
+
=
ax
by
cz
d
Both forms have their individual advantages, but the Cartesian form is useful from a geometric
perspective.
We are going to use vector analysis to derive the plane equation, and the reader will see
that there is an intimate relationship between this and the Cartesian form of the line equation
described in Section 3.3. Furthermore, we will approach the analysis in an identical fashion.
Step 1
Define a plane away from the origin O containing the point P x y z and its associated
position vector
p
, as shown in Fig. 4.1.
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