TABLE OF CONTENTS The mathematics and geometry of planes pervades computer graphics, geometric modeling, computer-aided design and manufacturing, scientific visualization, and many more applications. This chapter looks at various ways to define them, including the normal form and the three-point form, the relationship between a point and plane, and plane intersections.
Euclidean geometry defines a plane in space as the locus of points equidistant from two fixed points. The resulting plane is the perpendicular bisector of the line joining the two points. We call this definition a demonstrative or constructive definition. Computer-graphics and geometric-modeling applications require a more quantitative definition, because these planes may be bounded by polygons to form the polyhedral facets representing the surface of some modeled object or the plane of projection of some view of an object, and so on.
The classic algebraic definition of a plane is the first step toward quantification. The implicit Cartesian equation of a plane is
This is a linear equation in x, y and z. By assigning numerical values for each of the constant coefficients A, B, C, and D, we define a specific plane in space. It is an arbitrary plane because, depending on the values of the coefficients, we can give it any orientation. Figure 10.1 shows only that part of a plane that is in the positive x, y, and z octant. The three bounding lines are the intersections of the arbitrary plane with the principal planes.
