Hyperbolic regularization of the restricted three-body problem on curved spaces

We consider three bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature kappa. In this system, two bodies with equal masses form a relative equilibrium solution, these bodies are known as primaries, the remaining body of negligible mass does not affect the motion of the others. We show that the singularity due to binary collision between the negligible mass and the primaries can be regularized local and globally using hyperbolic functions. We show some numerical examples of orbits for the massless particle.