Geodesic Flow Computation on Parametric Surfaces via WebAssembly: A Differential Geometry Engine with Adaptive Numerical Integration

Presenting a web-based system for real-time computation and visualization of geodesic flows on parametric surfaces through direct numerical integration of the geodesic equation. The system implements complete differential geometry calculations including Christoffel symbols, Gaussian curvature, and mean curvature from first principles, compiled to WebAssembly for near-native performance. Using fourth-order Runge-Kutta integration with adaptive step sizing, the system achieves geodesic path computation in 10-30 milliseconds for 1000 integration steps. The WebGL-based visualization renders parametric surfaces with curvature-based color mapping at 60 frames per second on modern hardware. We validate the implementation against analytical solutions for spherical geodesics and demonstrate the system's capability across four distinct surface geometries: torus, sphere, hyperbolic paraboloid, and monkey saddle. The complete source code is available as open-source software, providing an educational tool for differential geometry and computational mathematics.