Line of intersection between two planes

Question

How can I find the line of intersection between two planes?

I know the mathematics idea, and I did the cross product between the the planes normal vectors

b

Answer 1

This method avoids division by zero as long as the two planes are not parallel.

If these are the planes:

A1*x + B1*y + C1*z + D1 = 0
A2*x + B2*y + C2*z + D2 = 0

1) Find a vector parallel to the line of intersection. This is also the normal of a 3rd plane which is perpendicular to the other two planes:

(A3,B3,C3) = (A1,B1,C1) cross (A2,B2,C2)

2) Form a system of 3 equations. These describe 3 planes which intersect at a point:

A1*x1 + B1*y1 + C1*z1 + D1 = 0
A2*x1 + B2*y1 + C2*z1 + D2 = 0
A3*x1 + B3*y1 + C3*z1 = 0

3) Solve them to find x1,y1,z1. This is a point on the line of intersection.

4) The parametric equations of the line of intersection are:

x = x1 + A3 * t
y = y1 + B3 * t
z = z1 + C3 * t