EddieOffermann dot com

Matrix math is a bitch. And it's nigh impossible to find any good resources online. Every "transform matrix tutorial" or "matrix multiplication lesson" online explains in detail how to multiply matrices together.

That hasn't been my problem for about 25 years now. I get it. This is how you multiply a matrix with a vector. This is how you multiply two matrices together. Neither of those things actually *accomplishes* anything if you don't know what the data in those matrices represents or what adding, subtracting or multiplying them would do.

My problems always center around something more like this: I have a point at <10,20,30> with orientation <15,30,45>. Where is the point if I rotate it 15 degrees in y then translate it 100 units in the X axis?

What's screwed up is that I've so frequently solved this by doing trig that I know off the top of my head that pi/180 is estimated at 57.29577951. Some people pride themselves on the number of digits they know pi to: I find the radian conversion number to be far more handy. But for reference, pi to me is 3.14159265. Note the 8 significant digits behind the decimal number: can you tell what decimal significance I grew up programming in?

The thing is, I'm working on a number of tools for managing stereo cameras in Maya. I need to quickly determine from the transforms of one camera (derived from a track of live footage) the location and orientation of a second camera if we know (or can estimate or otherwise determine) the interocular distance (the distance between the eyes) and the convergence (how far in front of these cameras their line of sight intersects). Determining the angle of the second camera relative to the first is a straightforward bit of trig: think Pythagoras and arctangents and it'll come to you, or go find a good trig reference online - this part's easily found in a million places. Diagram that out for yourself, making an equilateral triangle with cameras at each base corner with convergence being the length of a line bisecting the triangle into two right triangles. If it's not immediately clear, work on it a bit: it might wake up parts of your brain that haven't seen the light of day since highschool.

The tricky bit is figuring out where a right-eye camera would be if it was offset along a line that isn't quite perpendicular to the left-eye camera.

I'm very nearly there - based on a given camera, I can find a point offset along that camera's X axis and apply a convergence factor to the resulting camera location by canting it back in along the Y. But rotating a known transform matrix by an additional value in the y axis *before* doing that x translation... still fussing with that bit. I can make that happen less elegantly - by actually transforming nulls through space in Maya - but I'm searching for the magical transformation matrix that'll make it happen in a simple series of math statements.

It's late. It'll probably be plenty clear to me in the morning.

UPDATE: Got it! Sleep did wonders - my tired brain just didn't quite get the matrix arrangement right before.  I'll write an entry this weekend covering concatenated matrix transforms.  Damn handy.  Damn powerful.