This appendix is for the most part very straightforward. Nevertheless, there exists one sentence that really confuses me (marked bold in the quotation).
Spivak Calculus, page 84:
…In many situations it is more convenient to introduce polar coordinates, which are illustrated in Figure 1.
To the point $P$ we assign the polar coordinates $(r,\theta)$, where $r$ is the distance from the origin $O$ to $P$, and $\theta$ is the angle between the horizontal axis and the line from $O$ to $P$. This angle can be measured either in degrees or in radians (Chapter 15), but in either case $\theta$ is not determined unambiguously. For example, with degree measurement, points on the right side of the horizontal axis could have either $\theta=0$ or $\theta=360$; moreover $\theta$ is completely ambiguous at the origin $O$. So it is necessary to exclude some ray through the origin if we want to assign a unique pair $(r,\theta)$ to each point under consideration.
What is the meaning of the marked sentence? What ray is he talking about?
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 $(r,\theta)$ assigned to any of its points. And by rotating this ray it follows that no point in the plane can have a unique pair $(r,\theta)$ assigned to it, but isn't this already obvious at this point? I'm not so sure…
Best Answer
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.