Plane–plane intersection
In geometry, the intersection of two planes in three-dimensional space is a line or the empty set for parallel planes.
Formulation
The line of intersection between two planes and where are normalized is given bywhere
Derivation
This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product .The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as, since is a basis. We wish to find a point which is on both planes, so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for and.
If we further assume that and are orthonormal then the closest point on the line of intersection to the origin is. If that is not the case, then a more complex procedure must be used.