For many purposes we will need the formula for a plane, and this works as follows:
The plane may be of a uniform y type, in which case its formula is y = n.
If we wished to record the formula for this plane in, say, a simple array literal, we could write
"uniform y,n".
The uniform y type is a special case, and now we will move in the direction of the general case by saying that the uniform y plane never intersects the xz plane, while any non-uniform y plane does intersect the xz plane, and so, in the case of the non-uniform plane we want to define the intersection of the plane and the xz plane, which is a line.
In order to define this line we need some information about the plane, which will often be in the form of three points which are used to define the plane, which we can name A, B, and C, where a point N (for point name) is defined by the attributes N.x, N.y, and N.z. In the case of a uniform y plane
A.y = B.y = C.y = n. In the case of a non-uniform y plane A.y != B.y or B.y != C.y, in which case we can use a procedure to define the line of intersection with the xz plane, keeping in mind that now there is a second special case, the non-uniform y uniform x intercept plane. We will define the line on the xz plane using two points, which we can call D1 and D2, and if D1.x = D2.x the plane is of this second special type and its formula consists of a slope, change in y / change in x, and x = n = D1.x = D2.x, which is the formula for the intersection of the plane and the xz plane.
Finally, if the plane is of the non-uniform y type and D1.x != D2.x, the plane is of the general type and its formula combines the equation of the intercept line, so, z = mx + b, which we derive using a procedure, and a slope, change in y / change in z, when moving parallel to the z axis.
Thus if we name our plane types 1, 2, and 3, in the case of a plane of the first type its definition could be written "1,n" where n is a number and the formula for the plane is y = n. We can write the definition for the second type of plane "2,m,n", where m is the change in y / change in x slope and n is the value of the xz intercept, such that the equation for the xz intercept is x = n. Finally we can write the formula for the third type "3,m,b,my", where m is the slope of the xz intercept, b is the z intercept of that line, and my is the change in y / change in z slope in a direction parallel to the z axis.
