Hi all!
I'm using planes all the time and they work well for me, but if I'm honest
then I have to admit that I don't fully understand the plane equation
Ax+By+Cz+D=0. I've looked in all sorts of places for a good explanation
(including math books), but I wasn't able to find anything that really helps
me. Maybe someone can shed some light into it.
Here's what I *believe* I understand so far.
A plane is defined by the infinite set of points P(x, y, z) that satisfy the
condition Ax+By+Cz+D=0 (that is a plane is defined by the set of points that
lie on the plane). The coefficients A, B, and C are the components of the
vector normal to the plane, N = (A B C), and D is some constant. This
constant D seems to be the distance from the origin to the plane, if and
only if the normal vector N is a unit vector.
I also know that I can rewrite the above plane equation as (N _dot_ P) + D =
0. This is obvious, and I can use this information to classify a point
(determine in which halfspace it is) or to calculate the signed distance
from the plane to the point (if the normal vector N is a unit vector).
My question: "Why? Why does the above work the way it works?". I believe my
biggest problem here is that I have trouble to understand the true meaning
of the constant D.
Hope someone can help.
Niki
