I have a 3d plane at the world origin that is aligned with the world X/Y plane (facing the Z axis). I then have four 3d vertex positions for a new plane transformed into some location in 3d space.
Both planes have the same winding order for all 4 vertices.
I have a guarantee that the 4 corners are planar and there is no skewing (the plane may have still been scaled individually on the x/y axes).
How can I create a 4x4 transformation matrix given the final 4 corners of this plane?
Assume that the plane looks like this:
Construct a "local basis" of the plane, with the:
The transformation matrix can be decomposed into 3 components:
1 – Scale
Since the original quad has dimensions of 1x1 units, the scaling factor along the X and Y local axes are simply the side lengths, i.e. the lengths of AD and AB respectively. Ignore the Z scaling factor since the quad is planar.
Therefore the scaling component is given by:
2 - Rotation
The rotational component can be directly constructed from the local basis axes X, Y, Z; each vector (normalized) is the corresponding column of the matrix.
Therefore the rotational component is given by:
3 - Translation
This is the easiest one; the translation vector is simply the absolute coordinate of the quad's center O, and is equal to the last column of the matrix.
Therefore the translational component is given by:
The final matrix can be obtained by multiplying the above in the following order:
i.e. the components are applied in the order 1 ⇨ 2 ⇨ 3.
