Matrices can be your Friends.
By Steve Baker
Take an OpenGL matrix:
float m [ 16 ] ;
m[0] m[4] m[ 8] m[12] m[1] m[5] m[ 9] m[13] m[2] m[6] m[10] m[14] m[3] m[7] m[11] m[15]
...but we are OpenGL programmers - not mathematicians - right?! The reason OpenGL arrays are laid out in what some people would consider to be the opposite direction to mathematical convention is somewhat lost in the mists of time. However, it turns out to be a happy accident as we will see later.
If you are dealing with a matrix which only deals with rigid bodies (ie no scale, shear, squash, etc) then the last row (array elements 3,7,11 and 15) are always 0,0,0 and 1 respectively and so long as they always maintain those values, we can safely forget about them for now.
The first three elements of the rightmost column of the matrix is just the overall translation. If you imagine some kind of neat little compact object (like a teapot), then array elements 12,13 and 14 tell you where it is in the world. It doesn't matter what combinations of rotations and translations it took to produce the matrix, the rightmost column tells you where the object basically is. It is often fortunate that the OpenGL matrix array is laid out the way it is because it results in those three elements being consecutive in memory.
OK, so now we are down to only nine random-looking numbers. These are the top three elements of each of the first three columns - and collectively they represent the rotation of the object.
The easy way to decode those numbers is to imagine what happens to four points near to the origin after they are transformed by the matrix:
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