math - How to create 2d plot of arbitrary, coplanar 3d curve

Question

I have a set of points which comprise a (in theory) co-planar curve. My problem is that the plane is arbitrary and can move between each time I collect the data (these points are being collected from a camera). I was wondering if you guys could help me figure out how to:

1. find the plane which is closest to the one which these points are co-planar on
2. project the points on this plane in such a way that gives me a 2-d curve

I believe that I know how to do point 2, it is really mainly point 1 that i'm struggling with, but I wouldn't mind help on the second point as well.

Thanks a ton!

Answer 1

1. Find 3 points A,B,C in your data

   They must not be on single line and should be as far from each other as possible to improve accuracy.

2. Construct U,V basis vectors

    U = B-A
    V = C-A
   

   normalize

    U /= |U|
    V /= |V|
   

   make U,V perpendicular

    W = cross(U,V) // this will be near zero if A,B,C are on single line
    U = cross(V,W)
   

3. Convert your data to U,V plane

   simply for any point P=(x,y,z) in your data compute:

    x' = dot(U,P)
    y' = dot(V,P)
   

   in case you need also the reverse conversion:

    P = x'*U + y'*V
   

   In case you want/have an origin point A the conversions would be:

    x' = dot(U,P-A)
    y' = dot(V,P-A)
    P = A + x'*U + y'*V
   

   That will map A to (0,0) in your 2D coordinates.

In case you do not know your vector math look here:

• Understanding 4x4 homogenous transform matrices

at the bottom you will find the equation for vector operations. Hope that helps ...