# Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in \mathcal{S}$. For any real constant $0 \leq \theta \leq \pi$, define $$\mathcal{E}_\theta = \{P \in S\ |\ \angle AOP + \angle BOP = \theta\}.$$

$\mathcal{E}_\theta$ is a somewhat oval-shaped closed curve in $\mathcal{S}$, that is an analogue of an ellipse. Indeed, as the radius of $\mathcal{S}$ grows relative to $\theta$, $\mathcal{E}_\theta$ approaches the limiting case of an ellipse in a flat plane.

Similarly, by considering the difference in angle instead of the sum, we could define an analogue of hyperbolas.

Do these generalizations have interesting properties? Such analogues can be constructed in arbitrary metric spaces.

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