The point with Polar coordinates

Question

In spivak's calculus there is a section named Appendix 3 about polar coordinates which somewhere says that , it always make sense to talk about "the point with polar coordinates " , instead of "the polar coordinates "of a given point, and draws a graphs which say that if P is a point in third quadrant with polar coordinates (r,¤) then it has also polar coordinates (-r,●) where ¤ and ● represents angles made by P in 1st & 3rd quadrants counter clockwise. So my query is that how's that r could be negative & put some more light on this topic that how different convention of writing a point in polar coordinates means the same.

Answer 1

In polar coordinates we may use negative values for $r$ when it is more convenient.

For example if you wish to graph the polar function $$r=2-3\cos(\theta)$$ and you are given that $$0\le\theta \le \pi/2$$ you need to give values for $\theta$ and find values for $r$

If you start with $\theta =0$ you get $r=-1$ which is negative. The polar point $(-1,0)$ has also polar coordinates of $(1,\pi )$, but you are told to go with $0 \le\theta \le\pi/2$ thus $\pi$ is not allowed.