calculuspolar coordinates
From a calculus book I'm reading: "Unlike the graphs of an equation in $x$ and $y$, the graph of an equation $r=f(\theta)$ can be symmetric with respect to the polar axis, the line $\theta = \pi/2$, or the pole without satisfying one of the tests for symmetry. This is true because of the many different ways of specifying a point in polar coordinates."
Why is the previous paragraph true? Why are there many ways to specify a point in polar coordinates and how does that affect the symmetry test?
The spiral $ r=a\theta$ goes through origin. Rotate the full spiral as a rigid spiral by an angle $\beta. $
$ r= a(\theta + \beta) = a\theta + a \beta = a\theta + b$. The spiral has nonzero value where it cuts x-axis. Only after looking at $\beta$ in the opposite direction does the spiral goes through x-axis.
Did you draw the whole thing, or just the portion for positive $\theta$?
The usual Archimedean spiral, for $\theta$ from zero to $20$ (radians):
The spiral, for $\theta$ from $-20$ to $20$:
Do you see the symmetry now?
Most polar graphs you deal with, if you graphed them over a full period from zero to $2\pi$, wouldn't have this problem; the symmetry would show, if it's there. But the Archimedean spiral isn't periodic. In order for that symmetry to show up, we have to actually graph the parts with negative $\theta$.
Best Answer
André has posted what I think is a satisfactory answer in the comments, hence this wiki answer.