Modeling reflection and refraction of freeform surfaces

In this work, we present a detailed procedure of computer implementation of the laws of refraction and reflection on an arbitrary surface with rotational symmetry with respect to the propagation axis. The goal is to facilitate the understanding and application of these physical principles in a computational context. This enables students and instructors alike to develop simulations and interactive applications that faithfully replicate the behavior of light and sound propagating in a diversity of media separated by arbitrary surfaces. In particular it can help to explore freeform optics. Additionally, we include a practical example demonstrating these implementations using either Matlab or open-source Octave programming language.