In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. We teach the students to test for symmetry in standard ways:
- If the equation is invariant under the transformation $(r,\theta)\mapsto(r,-\theta)$, then we have symmetry across the polar axis. This is equivalent to the rectangular transformation $(x,y)\mapsto(x,-y)$.
- If the equation is invariant under the transformation $(r,\theta)\mapsto(r,\pi-\theta)$, then we have symmetry across the polar axis. This is equivalent to the rectangular transformation $(x,y)\mapsto(-x,y)$.
- If the equation is invariant under either of the the transformations $(r,\theta)\mapsto(r,\pi+\theta)$ or $(r,\theta)\mapsto(-r,\theta)$, then we have symmetry across the pole. This is equivalent to the rectangular transformation $(x,y)\mapsto(-x,-y)$.
These are all great, but then there was a homework question about the graph given by $r^2=\sin\theta$.
It's clear that we can replace $r$ with $-r$, so we have symmetry number 3. Additionally, $\sin\theta=\sin(\pi-\theta)$, we have symmetry number 2.
The function appears to fail the test for symmetry number 1, because $\sin(-\theta)=-\sin\theta$. However, any function with two of these symmetries automatically has the third. (Together with the identity transformation, we're just looking at the Klein 4-group here.)
Symmetry number 1 can be detected algebraically, but to do so, you have to use the equivalent transformation $(r,\theta)\mapsto(-r,\pi-\theta)$, which is clearly just the composition of symmetries 2 and 3. Similarly, symmetry number 2 could also be detected by the composition of 1 and 3: $(r,\theta)\mapsto(-r,-\theta)$
Anyway, the online WebAssign homework was only counting symmetries 2 and 3 right. I emailed them, and they say they've replaced the question, and I'll see the new version next time I make this assignment. Great, but that leaves math/teaching questions unanswered.
- Has anyone else had this issue come up, and how did you handle it?
- What is the easiest way to account for the failure of this equation, or one like it, when testing for symmetry number 1?
- If we applied two tests for each symmetry (one with $r$ and one with $-r$), are there any equations that would still "get away" undetected?
- Is it sufficient to state (A) that any function with two of these symmetries automatically has the third, and (B) any such function will pass two of the traditional tests, thus letting us know?
- Is there anything else students should be on the lookout for, when trying to obtain symmetries of polar graphs from analyzing equations in $r$ and $\theta$?