  Chaos, namely exponential sensitivity to initial conditions, is generally
considered a nuisance, inasmuch as it prevents long-term predictions in
physical systems. Here, we present an easily accessible approach to undo
deterministic chaos and tailor ray trajectories in arbitrary two-dimensional
optical billiards, by introducing spatially varying refractive index therein. A
new refractive index landscape is obtained by a conformal mapping, which makes
the trajectories of the chaotic billiard fully predictable and the billiard
fully integrable. Moreover, trajectory rectification can be pushed a step
further by relating chaotic billiards with non-Euclidean geometries. Two
examples are illustrated by projecting billiards built on a sphere as well as
the deformed spacetime outside a Schwarzschild black hole, which respectively
lead to all periodic orbits and spiraling trajectories in the resulting 2D
billiards/cavities. An implementation of our method is proposed, which enables
real-time control of chaos and could further contribute to a wealth of
potential applications in the domain of optical microcavities.
