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:
- find the plane which is closest to the one which these points are co-planar on
- 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!
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.
construct U,V basis vectors
U = B-A
V = C-A
normalize
U /= |U|
V /= |U|
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)
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 ...
