Is it possible to extract the model portion of a modelview matrix?

Question

(This is a similar question to extracting scale matrix from modelview matrix ) but I think it's a bit more general, so I'm reposting it.

I have a modelview matrix in WebGL, composed of a series of transformations from local object space to world space, followed by a transformation from world space to homogeneous WebGL coordinates (-1 to 1 in all directions).

This matrix is built up throughout our code over a series of convoluted steps. One of these steps is to render to a texture, and then blit the texture onto the screen. This process is extracted to be generalized so we can use it for any 2D texture painting operation. Unfortunately, it provides its own view/projection transformation. At the point at which I'm calling this, I have only the combined modelview matrix.

What I want to do is preserve the transformations made on the modelview matrix as it's built up along the way, but I don't want to include the view transformation. In other words, given a modelview matrix, and a known view transformation, is there a way to extract just the model transformation as a matrix?

We're not using perspective projection, and all of our transformations are 2-dimensional in nature, so a general solution isn't necessary (although I have had this question before when working in 3D, so something that could be extended to 3D would be really useful).

Answer 1

If i get it right you have Inverse(V)*M inside modelview and you are building it incrementally. That means in your code you are doing something like this:

glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
... here V transforms ...
... here M transforms ...
... here render or whatever ...

With multiple objects the stuff usually looks like this:

glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
... here V transforms ...
for (all objects)
 {
 glPushMatrix();
 ... here M transforms ...
 ... here render or whatever ...
 glPopMatrix();
 }

So to obtain V you can do this:

glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
... here V transforms ...
double iV[16],iVM[16];
glGetDoublev(GL_MODELVIEW_MATRIX,iV);
for (all objects)
 {
 glPushMatrix();
 ... here M transforms ...
 glGetDoublev(GL_MODELVIEW_MATRIX,iVM);
 ... here render or whatever ...
 glPopMatrix();
 }

now:

    iVM =   iV*M
V * iVM = V*iV*M
V * iVM =      M

where V = Inverse(iV) holds the direct view matrix and M holds the direct model matrix. As OpenGL does not have inverse matrix you can do it by transpose + position computation see:

• Matrix inverse accuracy

see the matrix_inv function in there so the result would be:

double M[16],V[16];
matrix_inv(V,iV);
matrix_mul(M,V,iVM);

Where matrix_mul code you can find in here:

• Understanding 4x4 homogenous transform matrices