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

but how to get the line from the resulted vector programmatically

Answer 1

The equation of the plane is ax + by + cz + d = 0, where (a,b,c) is the plane's normal, and d is the distance to the origin. This means that every point (x,y,z) that satisfies that equation is a member of the plane.

Given two planes:

P1: a1x + b1y + c1z + d1 = 0
P2: a2x + b2y + c2z + d2 = 0

The intersection between the two is the set of points that verifies both equations. To find points along this line, you can simply pick a value for x, any value, and then solve the equations for y and z.

y = (-c1z -a1x -d1) / b1
z = ((b2/b1)*(a1x+d1) -a2x -d2)/(c2 - c1*b2/b1)

If you make x=0, this gets simpler:

y = (-c1z -d1) / b1
z = ((b2/b1)*d1 -d2)/(c2 - c1*b2/b1)