Geodesic Curves Simulation on a Transcendental Surface with a Cycloidal Generatrix Using the Calculus of Variations

This article introduces a novel variational approach for solving the inverse geodesic problem on a transcendental surface shaped as a cylindrical structure with a cycloidal generatrix, a type of geometry that has not been previously studied in this context. Unlike classical models that rely on symmetric surfaces such as spheres or spheroids, this method formulates the geodesic path as a functional minimization problem. By applying the Euler–Lagrange equation, an analytical integration of the corresponding second-order differential equation is achieved, resulting in a parametric expression that satisfies boundary conditions. The effectiveness of the proposed method for computing geodesic curves on transcendental surfaces has been rigorously evaluated through a series of numerical experiments. Analytical validation has been carried out using MathCad, while simulation and three-dimensional visualization have been implemented in Python. Numerical experiments are conducted and 3D visualizations of the geodesic lines are presented for multiple point pairs on the surface, demonstrating the accuracy and computational efficiency of the proposed solution. This enables a closed-form analytical representation of the geodesic curve, significantly reducing computational complexity compared to existing numerical-heuristic methods. The obtained results offer clear advantages over existing studies in the field of computational geometry and variational calculus. Specifically, the proposed method enables the construction of geodesic curves on complex transcendental surfaces where traditional methods either fail or require intensive numerical approximation. The analytical integration of geodesic equations enhances both accuracy and performance, achieving an average computational cost reduction of approximately 27-30% and accuracy improvement of around 20% in comparison with previous models utilizing non-polynomial metrics. These enhancements are especially relevant in applications requiring real-time response and precision, such as robotics, CAD systems, computer graphics, and virtual environment simulation. The method’s ability to deliver compact and exact solutions for boundary value problems positions it as a valuable contribution for both theoretical and applied sciences.