Euler angles

And mathematically, you can use different parameterizations for this.

Specifically, the three common ones are Euler angles, or you could use what is called quaternions, or you could use the so-called direction cosines.

There's pros and cons to each of these in terms of mathematical complexity and, you know, intuitiveness and, you know, the gotchas that are tied to each of these in terms of, like, you know, where you need to be careful with how you do your math, we are going to use direction cosines for our analysis.

That's my favorite one.

Now let's go back to the projection matrix.

How many degrees of freedom we have? So how many things do we need to estimate in general? So if we go...

Yeah, here.

So K in general has, uh, four parameters: alpha, beta, U zero, V zero, sometimes skewness, so five.

R is a three by three matrix, but it's an orthogonal matrix, which means that, uh, uh, you have six, six constraints due to the orthogonality.

Which means that even though you have nine entries, you have three degrees of freedom, which correspond to the fact that, uh, you can, uh, reorient, um, coordinate system with respect to three, uh, basis angles, Euler angles.

And T is a three-dimensional as well.

So you have in total either six plus four or six plus five, so eleven degrees of freedom.