Why is the affine hull of the unit circle $\mathbb R^2$?

I guess I know your confusion. The point is that $\theta_i$ is not constrained to be greater than zero. So now you may understand why three non-collinear points will fill the whole $\Bbb R^2$.

To have an answer recorded as such, I'll add a few words. If $C$ contains the origin, then the affine hull is the same as linear span, since we can include $0$ with any coefficient we want. Also, translating $C$ by a vector translates its affine hull by the same vector. Thus, we can find the affine hull by moving the coordinate system so that the origin lies in $C$, and then taking the linear span. This shows at once that the affine hull of any three non-collinear points in the plane is the entire plane.

Take any point in $\mathbb R$. We can always draw a line through it which passes through two points in the circle. That means it lies on the same line that passes through those points on the unit circle.

That's the definition of Affine Hull. Please rectify me if I'm wrong.