Gaussian Curvature & Geodesics (2D)

This 2D companion drives the exact same differential-geometry math as the 3D version — analytic Gaussian curvature from a surface's first and second fundamental forms, and a geodesic integrated live via its Christoffel symbols — but reads it top-down instead of in relief: a bowl, a saddle, an alternating wave, or a flat plane rendered as a flat curvature heatmap, red where it curves like a dome and blue where it curves like a saddle. Launch a geodesic from any point at any angle and watch the yellow trace bend visibly whenever it crosses from a red region into a blue one, making the link between curvature sign and geodesic behavior easier to read than in a rotating 3D view.