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I did some tooling around to find an abelian group operation for the set of points on the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1$, given by $(p,q)*(r,s)=\frac{1}{a}(pr-\frac{a^2}{b^2}qs,ps+qr)$. I checked associativity, the identity is $(a,0)$, and the inverse of $(p,q)$ is $(p,-q)$.

What are some properties of this group worth exploring -- some good problems for someone with basic algebra experience?

Daniel Fischer
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This is the circle group for all non-zero choices of $a,b$. The map $$(x,y)\mapsto(\frac{x}{a}, \frac{y}{b})$$ is an isomorphism to the circle group. This group appears naturally if you want to see the effect of a rotated camera in an image if the camera has different resolutions in horizontal and vertical directions.

WimC
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