I have 4x4 matrix (A) (maya softw):
|Xx Xy Xz 0| - basis of X and dummy 0
|Yx Yy Yz 0| - basis of Y and dummy 0
|Zx Zy Zz 0| - basis of Z and dummy 0
|Tx Ty Tz 1| - Position x, y, z and dummy 1
In this case, I know there is no scaling or sheer, so the upper-left 3x3 portion is a pure rotation matrix. That means I can invert this portion by taking the transpose.
So inverting the matrix should give me something like this:
|Xx Yx Zx 0|
|Xy Yy Zy 0|
|Xz Yz Zz 0|
|T? T? T? 1|
But I don't know how to find the T?
values that correctly invert the translation component of the matrix. It's not just the inverted position (T-1)
My thinking so far:
Matrix of an untransformed object at the origin - (I - identity):
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
Matrix of a translated object (Tx=3, Ty=4, Tz=5) (A):
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 3 4 5 1 |
Inverse matrix of A (A'):
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
|-3 -4 -5 1 |
Matrix of translated and rotated(Y=90) object(B):
| 0 0 -1 0 |
| 0 1 0 0 |
| 1 0 0 0 |
| 3 4 5 1 |
inverse matrix of B (B'):
| 0 0 1 0 |
| 0 1 0 0 |
|-1 0 0 0 |
| 5 -4 -3 1 |
Strange things start to happen when you take two differently-rotated matrices with the same translation and you get different inverse translations!
It seems like some dot product is involved in the translation calculation.