Well, for 2D, you could do it with a single number - the angle. :)

But if you want to avoid trigonometric functions, it’s certainly possible to do it by storing only the first row of the matrix, reconstructing the second row on demand. This is sort of the 2D analogue of using quaternions instead of 3x3 rotation matrices. In fact, the first row of a 2D rotation matrix could be thought of as real and imaginary parts of a normalized complex number, which represents rotation in 2D in the same way that quaternions represent rotation in 3D.

A 2D matrix that contains only a rotation cannot have a form other than (cos(a), sin(a), -sin(a), cos(a)) and that means I should only be storing 2 numbers - the other two are implicit. Not only does that improve my chances of hitting the cache with complex data objects, it means there are two less places for rounding errors to mess everything up. So, my 2D physics engine will now be storing orientations as two floats instead of four.

Is this:

a) Neat

b) Old hat

c) Wrong