# How do I find the generalized coordinates in a certain system?

I'm learning about constraints and I know the following: If there are $N$ particles in 3 dimensional space, I have $3N$ degrees of freedom. If I have $n_b$ holonomic constraints and I switch over to generalized coordinates, I'm left with $n_q = 3N - n_b$ degrees of freedom.

Now let me give an example:

1 particle moves over the surface of a sphere. This initially gives me 3 degrees of freedom ($x$, $y$ and $z$). Now using the 1 holonomic constraint I have (the surface of the sphere, $x^2+y^2+z^2=R$) I should be left with 2 degrees of freedom.

This is where I get lost and don't know what else to do, my book says these 2 new degrees of freedom are angles, namely $\theta$ and $\phi$ and that that would give me the following:

$$x=R\sin{\theta}\cos{\phi}$$
$$y=R\sin{\theta}\sin{\phi}$$ $$z=R\cos{\theta}$$

How do I know that these 2 new degrees of freedom (my generalized coordinates) are angles and how do I find the equations to express $x$, $y$ and $z$ in these new coordinates?

For your particular example, in addition to the given parametrization $$(\phi,\theta)\mapsto(R\sin\theta\cos\phi,R\sin\theta\sin\phi,R\cos\theta)$$ we could also use $$(z,\phi)\mapsto(\sqrt{R^2-z^2}\cos\phi,\sqrt{R^2-z^2}\sin\phi,z)$$ inspired by cylindrical coordinates, and stereographic projection would work equally well.
In fact, the upper hemisphere can be covered with just $$(x,y)\mapsto(x,y,\sqrt{R^2-x^2-y^2})$$ ie one way to get a parametrization is by 'inverting' the constraints after a choice of independent variables.