Disk of the hyperbolic plane forced into Euclidean space, showing characteristic saddle ruffling
A disk-shaped region of the hyperbolic plane is meshed in polar coordinates and embedded into R3 using a spring-and-charge network. Spring rest lengths are set to match the hyperbolic metric ds2=dr2+sinh2(r)dθ2, so that the intrinsic geometry of the mesh approximates constant negative curvature K=−1.
The embedding is computed by a damped dynamics simulation with shear and bending springs for stability and Coulomb repulsion between vertices to prevent self-intersection. The resulting surface exhibits the saddle-like ruffling characteristic of hyperbolic geometry embedded in Euclidean space.
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