[Solved] Coordinate system transformation, 3d projection to 2d plane

Question

The problem can be expressed as finding the 3-by-3 matrix M such that the coordinates of a point P can be converted between the old coordinate system (P_old, 3 rows) and the new coordinate system (P_new, 3 rows). This is an affine transformation:

P_old = Center + M * P_new     (1)

The (matrix-vector) multiplication with M orientates it back to the old system, and adding Center‘s coordinates translates it back to the old origin.

The equation (1) can then be turned into:

P_new = M^{-1} * (P_old - Center)     (2)

where M^{-1} is the inverse of M, to compute the new coordinates from the old ones (the third row will have a 0 if the point belongs to the plane of the triangle).

The matrix M is made of the coordinates of the new basis in the old system, one basis vector in each column. One must now find such a basis.

This basis can be taken from (this is all pseudo-code):

Renormalizing AB
```
       AB
V1 = ______
    || AB ||
```
- AB here is meant as the vector AB (with an arrow on top):
```
|b_x - a_x|
|b_y - a_y|
|b_z - a_z|
```
- || . || is the Euclidian norm (^2 means the square, not xor):
```
|| V || = sqrt(V_x^2 + V_y^2 + V_z^2)
```
AC (also a vector, defined like AB), but minus its projection on V1 to make it orthogonal to V1, and renormalized (this will fail with a division by zero if the triangle is not really a triangle):
```
        AC - (AC.V_1) V1
V2 = _______________________
     || AC - (AC.V_1) V1 ||
```
- M.N is the scalar product:
```
M.N = M_x * N_x + M_y * N_y + M_z * N_z
```
- (AC.V_1) V1 is simply the product of a scalar, (AC.V_1), with a vector, V1

A third vector that can be taken as the cross product to get a Cartesian coordinate system:

V3 = V1 x V2

The cross product is defined as:

          |V1_y*V2_z - V1_z*V2_y|
V1 x V2 = |V1_z*V2_x - V1_x*V2_z|
          |V1_x*V2_y - V1_y*V2_x|

Then M can be taken as |V1 V2 V3| (each Vx is on 3 rows) and, then its inverse computed to use formula (2).

This transformation (with the inverted M) should both generate new coordinates for the points on the plane of the triangle that have 0 on the third axis (which makes it 2D-coordinates on that plane), and preserve the size in terms of Euclidian norm.

Accepted Answer