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Wednesday February 1, 2023
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The Maths of 3D

You can't have 3D without a little maths...


Line with Plane Intersection Tutorial

by bkenwright@xbdev.net







We'll brake up our solution finding, into simple steps....its the way that I learned it, and I think its easier to follow this way.


o Step 1.

Equation of our Line

         P(t) = P0 + t(P1 - P0)


This equation is also popularly known as a parametric line equation.


    P0 is the starting point of our line

    P1 is the end point of our line

    t is variable which goes from 0 to 1, and is used to derive a position at any point along our line.


o Step 2.

Equation for our Plane

       N dot (Px - P2) = 0



    Px is the point of intersection of our line

    P2 is any point on the plane

    N the normal for our plane

This equation is quit logical when you think about it, as when N and (Px-P2) are at 90 degrees to each other.  Remember (Px-P2) is the vector from P2 to our intersection point.


o Step 3.

Substitute 1 into 2.


       N dot (Px - P2) = 0

       N dot { P0+t(P1-P0) - P2 } = 0

       N dot { P0+t(P1-P0) } - N dot (P2) = 0

       N dot { P0+t(P1-P0) } = N dot (P2)

       N dot (P0) + N dot { t(P1-P0) } = N dot (P2)

       N dot { t(P1-P0) } = N dot { P2-P0 }


 t is a constant so


      t * N dot { P1-P0 } = N dot { P2-P0 }


      t =  N dot {P2-P0} /  N dot {P1-P0}



o Step 4.

We can substitute t back into our Line Equation to find the Point of intersection Px.


     P(t) = P0 + t(P1 - P0)


     Px = P0   +   [ N dot {P2-P0} /  N dot {P1-P0}]  *   (P1 - P0)






We should test for our line being perpendicular to the plane at the start, which usually means testing if  Ndot(P1-P0)==0 , which tells us that our line is never going to cross our plane.


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 1st end point.






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