The exponential mapping creates a metric of the form
\begin{align*}
ds^2=dr^2+\psi(r,\theta)^2d\theta^2,
\end{align*}
locally in a sufficiently small neighbourhood of any point $p$ on a two-dimensional manifold. Your condition expresses a local symmetry: on a sufficiently small neighbourhood of $p$ the metric is invariant with respect to a translation in the $\theta$ coordinate (we could call it a 'local rotation').
It makes a lot of difference if you want that kind of metric in one point or in every point, and in the latter case if the function $\psi$ has to be the same for every point (it isn't in your examples with $\psi^2(r)=r,r^3\ldots$).
If the same function $\psi(r)$ is valid around every point $p$ then we are in the realm of surfaces of constant curvature; to see this, note that the curvature for a general $\psi(r)$ is given by
$$K=-\frac1{\sqrt G}\frac{\partial^2\sqrt G}{\partial r^2}$$
where $G=1.\psi^2(r)$ is the determinant of the matrix expressing the components of the metric tensor. This shows that the function $\psi(r)$ uniquely determines the curvature at the origin of the coordinate system. If all origins have the same function $\psi(r)$ then all origins have the same scalar curvature.
Thus constant curvature implies $\psi''/\psi$ must be constant. Depending on the sign of the constant (the opposite of the sign of the curvature), the solutions of that differential equation are: first-degree polynomials (your example of a Euclidean plane), a scaled, shifted version of the hyperbolic sine (your example of a hyperbolic sphere) and a scaled, shifted version of the ordinary sine (a sphere). The shift can be eliminated by requiring $\psi(0)=0.$
The sphere of radius $R$ has $\psi(r)=\sin(r/R).$
I think this is supposed to be an example of one ray (that contains all points on the right side of the horizontal axis and the origin) which cannot have a unique pair (𝑟,𝜃) assigned to any of its points.
Yes, I think this is what he means. I was confused by this paragraph as well.
He doesn't really say it, but of course our choice of $\theta$ is somewhat arbitrary, so long as it covers the full 360°-$2\pi$ radians.
We can use $\theta$'s defined by
$$0 \leq \theta \leq 360$$
or
$$73 \leq \theta \leq 433$$
or
$$-\frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}$$
(Analogously, we could choose any ray through the origin against which we measure $\theta$. The choice to use the positive $x$-axis is somewhat arbitrary.)
In general we can use
$$\theta_0 \leq \theta \leq\theta_0 + 360$$
where $\theta_0$ is any initial angle.
I think what he's getting at with his "exclude some ray through the origin"
is that we exclude all points along the ray defined by $(r,\theta_0)$ where $r \geq 0$.
For any choice of $\theta_0$, we exclude the points $(r,\theta_0)$, along with the origin, leaving us only with points that can be unambiguously assigned to unique $(r, \theta)$, where $\theta_0 < \theta < \theta_0 +360$ and $r > 0$.
Analogously, We can chose to measure our $\theta$ against the positive $y$-axis, or the ray that's 76° counterclockwise from the negative $x$-axis, or any other ray through the origin, but regardless of which one we chose we must exclude points along this ray including the origin if we want to be able to assign unique $(r, \theta)$ coordinates.
This of course is not really satisfactory and will leave us with points we can't describe regardless of our choice of $\theta_0$
We could instead, restrict our $\theta$ to either
$$\theta_0 < \theta \leq \theta_0 + 360$$
or
$$\theta_0 \leq \theta < \theta_0 + 360$$
and define the origin point as having some fixed $\theta$, for example $\theta_0$ or $\theta_0 + 360$, depending on our choice of allowed $\theta$. We can easily avoid the ambiguity Spivak's worried about, without excluding any rays through the origin.
Fortunately, it's not really important. He goes on to say that, despite any difficulties in picking points on the plane and then assigning to them unique $(r, \theta)$ coordinates, going the other way is no problem at all.
Given any $(r, \theta)$, this maps to a unique $(x,y)$ in the plane, and the rest of the chapter involves this process.
Best Answer
Yes, in my opinion one can indeed define "polar coordinates" for an evenly spaced grid. Of course, things will be a bit different in this geometry. You have chosen to define $r$ as follows:
$$r = |l| + |n|$$
We now seek a meaningful angle $\theta$. In order to use Pythagoras, we first define the Euclidic norm:
$$E = \sqrt{l^2 + n^2}$$
We can now write $l = E \cos (\theta)$ and $n = E \sin (\theta)$. Combining these equations we arrive at the following expressions, valid for the first quadrant:
$$\cos (\theta) = l/E = \frac {l} {\sqrt {l^2 + (r-l)^2} } $$
$$\sin (\theta) = n/E = \frac {r-l} {\sqrt {l^2 + (r-l)^2} } $$
We see that the allowed values for $\theta$ unfortunately (but not surprisingly!) depend on $r$.