Line-Plane Intersection

STEP 1 : KNOWNS

The equation of a line is:

P = P0 + t(P1 – P0)

- P  is any arbitrary point on the line.  In this case, P is the point of intersection of the ray and the plane

- P0 is the starting point of the line

- P1 is the end point of our line segment

- t is the parameter.  Values between 0 and 1 represent points on our line.

The equation of a plane is:

N dot (P – P2) = 0 

- N is the normal of the plane

- P and P2 are known points on the plane.  In this case, P will represent the point of intersection. If our plane was represented by a triangle, P2 could be one of the vertices.

STEP 2 : PLUG AND CHUG

Plug the equation of the ray into our intersection pt of the equation of the plane.

N dot ( (P0 + tV) – P2) = 0

Solve for t:

(N dot (P0 + tV)) – (N dot P2) = 0

(N dot (P0 + tV)) = (N dot P2)

(N dot P0) + (N dot tV) = (N dot P2)

N dot tV = (N dot P2) - (N dot p0)

t = N dot (p2 – p0)

     N dot V

Note: If N dot V = 0, then the line segment is perpendicular to the Normal of the plane, meaning the plane and line-segment are parallel, no intersection.

STEP 3 : FINISHING UP

Now that you have t, plug it into our equation for a line:

P = P0 + t(P1 – P0)

And you’ll get the coordinates of the intersection point

Notes on T:

If:

t > 0 and t < 1  :           The intersection occurs between the two end points

t = 0                 :           The intersection falls on the first end point

t = 1                 :           Intersection falls on the second end point

t > 1                 :           Intersection occurs beyond second end Point

t < 0                 :           Intersection happens before 1^st end point.