Intersection of three planes

From CGAFaq

Three planes whose normals are not coplanar will intersect at exactly one point.

The three planes can be written as:

where is a unit normal for plane , is the displacement of the plane from the origin, and is an arbitrary point.

The unique intersection point , if it exists, is given by:

The expression in the denominator is called the triple product of the three normals, and is equivalent to the determinant of a 3×3 matrix with those columns. It is zero precisely when the vectors are not linearly independent, meaning the normals are coplanar. In this case the planes may intersect in a common line, or not at all.

As a convenient test, the point is the intersection of three axis-aligned planes: