Geometric Primitives - 3D Math Primer for Graphics and Game Development - page 312

9.5.1 The Plane Equation: An Implicit Definition of a Plane

We can represent planes using techniques similar to the ones we used to
describe infinite 2D lines in Section 9.2.2. The implicit form of a plane is
given by all points p = (x,y,z) that satisfy the plane equation:

ax + by + cz = d (scalar notation),

The plane equation

p n = d (vector notation). (9.10)

Note that in the vector form, n = [a,b,c]. Once we know n , we can compute
d from any point known to be in the plane.

Most sources give the plane equation as ax + by + cz + d = 0. This has the
effect of flipping the sign of d. Our comments in Section 9.2.2 explaining
our preference to put d on the left side of the equals sign also apply here:
our experience is that this form results in fewer terms and minus signs and
a more intuitive geometric interpretation for d.

The vector n is called the plane normal because it is perpendicular
(normal) to the plane. Although n is often normalized to unit length, this
is not strictly necessary. We use a hat ( n ) when we are assuming unit
length. The normal determines the orientation of the plane; d defines its
position. More specifically, it determines the signed distance to the plane
from the origin, measured in the direction of the normal. Increasing d slides
the plane forward, in the direction of the normal. If d > 0, the origin is
on the back side of the plane, and if d < 0, the origin is on the front side.
(This assumes we put d on the right-hand side of the equals sign, as in
Equation (9.10). The standard homogenous form with d on the left has the
opposite sign conventions.)
Let's verify that n is perpendicular to the plane. Assume p and q
are arbitrary points in the plane, and therefore satisfy the plane equation.
Substituting p and q into Equation (9.10), we get

n p = d,
n q = d,
n p = n q ,
n p − n q = 0,
n ( p − q ) = 0. (9.11)