From theory to visualization: deriving and simulating catacaustic curves

This article investigates the methodologies for calculating the caustics of plane curves, focusing on both parametric curves and algebraic curves defined by bivariate polynomials. We have derived the specific equations for caustics in these cases through theoretical analysis and validated them using the Python Sympy symbolic computation library. Additionally, simulation experiments were carried out using the POV-Ray renderer. Our research not only offers a combined theoretical and practical approach to study and compute caustics but also highlights the beauty of mathematics through specific examples, providing a new perspective for the mathematical modeling and simulation of optical phenomena.