Gaussian Curvature & Geodesics

Differential geometry studies how a surface bends in space and what "straight" means once you're confined to it. This simulation renders a deformable parametric surface — a bowl, a saddle, an alternating wave, or a flat plane — colored point-by-point by its Gaussian curvature K, computed analytically from the surface's first and second fundamental forms. Launch a geodesic from any point at any angle and watch it integrated live via the surface's Christoffel symbols: on the bowl it curves back on itself like a great circle on a sphere, on the saddle it fans outward, and on the flat plane it stays a straight line, exactly as the sign of K predicts.