Hey guys!

Quite simply, I was wondering if I maintain a transformation matrix of translations, scalings and rotations and also another one with just the rotations - could I extract the correct resulting scale and possibly translation if I multiply the combined matrix by the inverse rotation matrix? I have a feeling the math might not be that simple, but I'm just not sure. Thanks in advance!

Please look at my post here since it handles simple decomposition as well as its requisites.

EDIT: To talk about your approach:

When using row vectors, the combined matrix written as composition looks like

S * R * T

If you multiply with the inverse rotation on the local side you get

R-1 * S * R * T

Thanks haegarr! That is a beautifully comprehensive explanation. So I gather that in order for an inverse transformation to have a directly inverse effect, it had best be applied right next to the original transformation, on either side like R-1 * R * S * T or S * T * R * R-1?

As long as you don't want to interpret it w.r.t. extra-ordinary rotation center and/or scale axes and center, this means to have the equivalence

I am just a bit unclear about this part. Is the transformation application order important for this decomposition method to work?

[...]So I gather that in order for an inverse transformation to have a directly inverse effect, it had best be applied right next to the original transformation, on either side like R-1 * R * S * T or S * T * R * R-1?

Yes, because for any matrix M

M * M-1 = M-1 * M = I

and

M * I = I * M = M

so you have e.g.

R-1 * R * S * T = ( R-1 * R ) * S * T = I * S * T = S * T

However, notice that R * S * T is an order one usually don't want to use, because scaling appears in the rotated (if using row vectors) space, so that the axes of scaling are not what one probably expects, or else scaling appears in the translated (if column vectors are used) space, what means that the center of scaling is not where one probably expects. See below for details.

I am just a bit unclear about this part. Is the transformation application order important for this decomposition method to work?

What I meant here is that the composed matrix is defined to be (still using row vectors)

S * R * T

what means that center and axes of scaling and center of rotation is not freely available.

You can use other compositions, too. For example in X3D a transform is defined as

C-1 * A-1 * S * A * R * C * T

where C denotes the center of scaling and rotation, and A denotes the axes (in form of a rotation) of scaling. Decomposing such a matrix means to not only look for S, R, and T, but also for C and A and hence is more complex than what I have shown.

Fantastic! I'm currently using SRT-transformed mesh hierarchies in a game, and this should help me extract the information needed to form matching collision volumes. You have my gratitude :)

[... ]I'm currently using SRT-transformed mesh hierarchies in a game, and this should help me extract the information needed to form matching collision volumes. [...]

Notice that the transform definition you used originally to calculate the overall matrix plays no role here. The both compositing schemes I've shown above are to be understood as prescriptions for the de-composition, i.e. which matrices occur with what meaning. You can trouble-freely build a matrix by SRT and decompose it using the X3D scheme and vice-versa.

Can someone explain to me why we need non-uniform scalings at all?

Perhaps I don't understand your point. Non-uniform scaling works nicely for objects that "grow" more in one direction than another - e.g., a circus balloon that, during inflation, grows longer at a greater rate than the diameter. EDIT: Or Pinocchio's nose