Polar coordinates of a set with different radius and angle

Question

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<r< r_0,\, 0<\theta<\theta_0 \rbrace}$$ and there exist also polar coordinates with center $\tilde{q}\neq q$ such that $$U=\lbrace{ (r,\theta): 0<r< \tilde{r}_0,\, 0<\theta<\tilde{\theta}_0 \rbrace}.$$

I am wondering if it is possible to have $r_0 \neq \tilde{r}_0$ or $\theta_0 \neq \tilde{\theta}_0$? Or is there a unique radius and angle for each set ?

Answer 1

The angles $\theta$ and $\tilde\theta$ might be different.

Imagine the surface of revolution of some a graph $y=f(x)$, where the function $f\colon [0,1]$ is concave, $f(0)=f(1)=0$ and $|f'(0)|\ne |f'(1)|$. A slice along the meridians between the poles $q=(0,0,0)$ and $\tilde q=(1,0,0)$ has different angles $\theta$ and $\tilde\theta$.

Assuming $f''(0)=f''(1)=0$, this slice can be part of nice surface.