Gaussian Curvature & Geodesics

Surface

Curvature strength 0.22

Wave frequency 1.2

Geodesic launch

Start position (-1.8, -1.8)

Launch angle 35°

Integration speed 1.0x

K at start point

–

Surface type

–

Path length

–

Status

idle

How it works

The surface is the graph of a height function y = f(u,v) over a flat (u,v) domain. Its first fundamental form is E = 1+f_u², F = f_uf_v, G = 1+f_v², and its Gaussian curvature comes straight from the second derivatives of f:

L = f_uu/√(1+f_u²+f_v²)   M = f_uv/√(1+f_u²+f_v²)   N = f_vv/√(1+f_u²+f_v²)
K = (LN − M²) / (EG − F²)

Red patches are elliptic (K>0, dome-like — geodesics converge, like great circles on a sphere); blue patches are hyperbolic (K<0, saddle-like — geodesics fan apart); white is flat (K≈0).

A geodesic is "locally straight" on the surface: it satisfies the geodesic equation using Christoffel symbols Γ built from E,F,G and their u,v-derivatives:

u'' + Γ¹₁₁u'² + 2Γ¹₁₂u'v' + Γ¹₂₂v'² = 0
v'' + Γ²₁₁u'² + 2Γ²₁₂u'v' + Γ²₂₂v'² = 0

Curvature strength — scales the height function, so it directly scales K.
Wave frequency — only affects the Wave surface, whose curvature alternates sign across the grid (an "egg-carton" landscape).
Start position / Launch angle — the geodesic's initial point and direction in the (u,v) domain; the equations above are integrated forward in real time with 4th-order Runge–Kutta.

Watch how a geodesic launched straight bends visibly toward or away from itself purely from the surface's own curvature — no external force is applied, exactly as Gauss's Theorema Egregium says curvature is intrinsic to the surface.

K > 0 (elliptic) K ≈ 0 (flat) K < 0 (hyperbolic) Geodesic marker

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