Convert points on a 3d Plane to 2d Coordinates

Question

I have a point cloud, all the points lie on a plane in 3D Space. I need to convert each point to 2D Coordinates and vice versa.

(x,y,z) in Coordinate System A => Transformation Matrix (T1) => (x,y) in Coordinate System B

(x,y) in Coordinate System B => Transformation Matrix (T2) => (x,y,z) in Coordinate System A

I need T1 and T2. The coordinate system B can be any arbitrary reference frame.

Answer 1

As far as I understand, all points lie in the same plane, and you want to reduce dimension and later restore coordinates.

Get three non-collinear points A, B, C. Make vectors AB and AC.
Normal to that plane is

 N = AB x AC //cross product

Now normalize vectors AB and N getting unit U = uAB and uN. Build the second base vector (it is unit and lies in the plane)

 V = U x uN

Now you have four basis points A, u=A+U, v=A+V, n=A+uN

Tranformation should map these points into quadruplet (0,0,0), (1,0,0), (0,1,0), (0,0,1) correspondingly.

Now about affine transformation matrix to make this mapping:

      [Ax ux vx nx]   [0 1 0 0]
 M *  [Ay uy vy ny] = [0 0 1 0]
      [Az uz vz nz]   [0 0 0 1]
      [1  1  1  1 ]   [1 1 1 1]

or

  M * S = D
  M * S * Sinv = D * Sinv
  M = D * Sinv

So calculate inverse matrix for S=[Ax ux...] and get needed matrix M.

Application of M to any point in the plane gives new coordinates with zero z-component.

Application of inverse of M to (x,y,0) results 3D coordinates in given plane.

---

Maple sheet with points A=1,1,1 B=2,1,1 C=1,1,2 (in plane Y=1)

new coordinates AA, BB, CC have zero z-component.

For arbitrary point in the same plane z-component after mapping is zero too.

 P:=vector([-2,1,7,1]);
 > PP := multiply(M, P);
 PP := [-3, 6, 0, 1]