Intersection Of Two Planes

From CGAFaq

How do I find the intersection of a hyperplane and a 2-flat?

For the following we call the 2-flat a plane.

Parametrize plane A with (u, v):

where

• is a point on plane A
• and are non-parallel (positive length) tangent vectors in the

plane A

All points lie on plane A. We wish to search for a set of points that also lie on hyperplane B. We make the old trick of specifying the hyperplane as an implicit equation:

where

• is a point on the hyperplane B
• is the normal of the hyperplane B

Implicit equations and parametric equations mix nicely, as we can see by substituting the first equation to the second:

By bilinearity of dot product we can separate the u and v out:

Solving for u yields:

Here we assumed . Geometrically this means that planes A and B are parallel, in which case there is no intersection or the planes are the same.

Substituting u back into A(u, v) gives:

Rearrange to separate v:

To shorten notation let:

Then:

Which is a ray parametrized w.r.t v.

Note: