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Manifolds & Parallel Transport Simulator

Interactive differential-manifold simulator: pick a point on a sphere, torus, saddle or plane, build its tangent plane from real partial derivatives, then watch a vector parallel-transported around a closed loop return rotated by the Gaussian curvature's holonomy angle.

Mathematics3DAdvanced60 FPS

Details

CategoryMathematics

RenderingThree.js · WebGL

DifficultyAdvanced

Target FPS60

ControlsSurface buttons (Sphere / Torus / Saddle / Plane) switch the 2-manifold; click anywhere on the surface to place a point and build its tangent plane, basis vectors and normal; loop-radius slider sets the size of the parallel-transport contour; Parallel-transport button animates a vector once around the loop; Reset clears the selected point.

Three.jsdifferential geometrymanifoldsparallel transportGaussian curvatureholonomy

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⚙ Under the hood

Pick a point on a sphere, torus, saddle or plane and build its tangent plane from the surface's exact partial derivatives, then parallel-transport a vector around a closed loop to see holonomy: the vector returns rotated by an angle proportional to the enclosed area times the Gaussian curvature.

Three.jsdifferential geometrymanifoldsparallel transportGaussian curvatureholonomy

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install