The normal process for deriving a formula for a plane begins with the derivation of the equation for a line which is the intersection of the plane and what we call an axial plane.
Our plane exists in three dimensions which are defined by three axes, the x axis, the y axis, and the z axis. These three axes define three axial planes. As stated, the normal process derives the formula for a line that is the intersection of the plane we seek to describe and one of the axial planes, and, since we are speaking of where to begin a process, the x and y axes are in a sense native, because they are so familiar from two dimensional geometry. We will begin, then, by attempting to derive the formula for the intersection of or plane with the plane defined by the x and y axes, the x - y axial plane ... if, indeed, the plane does intersect the x - y plane.
So, we can begin by determining whether the plane indeed does so. This is a simple enough procedure.
Given three points defining a plane, (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3):
If (z3 != z1 or z2 != z1) {P does intersect x-y}. The symbol != means "not equal to", or "does not equal."
A variation is: if (z3 = z1 and z2 = z1) {the formula for the plane is z = z1} else
{find the intersection of P and x - y}.
