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Thursday February 2, 2023
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The Maths of 3D

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


Minimum Distance From A Point To A Plane Tutorial

by bkenwright@xbdev.net






Both methods generate the same solution, but its how you think about solving it....one method uses a parametric line view of it, the other things more in terms of planes and the plane equation.


...Using Method 1....


o Step 1.


Lets make some assumptions first, n is our plane normal, and its normalised.  So |n|=1. 


    P1 to P0 = P1P0 = P0-P1


And P1 is a point on our plane.

d is the shortest distance from our point P0 to the plane


o Step 2.



This gives us:

    d = n dot |P0 - P1|


o Step 3.


Now d is a distance, its a value, not a vector.  So we still need to determine the point Px on the plane.  We use our n direction value to know which way where going, and combine it with an amplitude so we know how far to go.  (Not forgetting P0 as our starting point)


    Px = P0 + d*n



    d = n dot |P0-P1|



....Method 2....

For this second method, we sort of work with a non-normalised plane vector and the plane equation.



As before, our point P0, and we want to find the closest distance to the plane.  Our plane is defined by P1, a ponit on the plane (x,y,z) and the planes normal n.

Now for a plane, any point on the plane must satisfy the plane equation, which is:


A x + B y + C z + D = 0


The minimum distance is then the absolute value of:


   ( A x0 + B y0 + C z0 + D ) / sqrt( A2 + B2 + C2 )



But ackk...what is that D value?


D is the distance from the plane to the origin.


    minimum distance = d = (A (x0 - x1) + B (y0 - y1) + C (z0 - z1)) / sqrt(A2 + B2 + C2)

                                      = ( A x0 + B y0 + C z0 + D ) / sqrt(A2+B2+C2)

                                      = [  (n dot P0) - (n dot P1) ] / sqrt(A2+B2+C2)

                                      = n dot (P0 - P1) /  sqrt(A2+B2+C2)



We don't have a normalised normal n for the second method example, which is why we divide by |n|.








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