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〰️ Gaussian Curvature Explorer

Interactive Gaussian curvature explorer. Compare a sphere, plane, saddle, cylinder and torus rendered as rotatable colour-coded meshes, with the real differential-geometry curvature formula computed at every point of each surface.

Geometry3DAdvanced60 FPS

〰️ What It Demonstrates

Gaussian curvature K measures how a surface bends at each point by multiplying its two principal curvatures (the maximum and minimum bending in perpendicular directions). A sphere bends the same way in every direction, giving constant positive curvature K = 1/R². A saddle bends oppositely along its two axes — up one way, down the other — giving negative curvature. A flat plane and, surprisingly, a cylinder both have K = 0: the cylinder curves strongly in one direction but not at all along its length, and since K multiplies the two principal curvatures, one being zero makes the product zero even though the surface clearly looks "curved" to the eye.

That cylinder example reveals the deep distinction between intrinsic and extrinsic curvature. Gaussian curvature is intrinsic — a two-dimensional being living on the surface could measure it purely by measuring distances and angles within the surface, without any notion of the 3D space it sits in. This is exactly why you can roll a flat sheet of paper into a cylinder without stretching it (both have K = 0) but can never flatten a sphere without tearing or distorting it (K ≠ 0 for a sphere, K = 0 for a plane — they are intrinsically different).

How to Use

Select a Surface: sphere, plane, saddle, cylinder or torus.
Drag the canvas to rotate, or enable Auto-rotate with adjustable Speed.
Increase Resolution for a finer mesh.
For the torus, move the r/R ratio slider and watch the blue (negative curvature) region on the inner rim grow or shrink relative to the red (positive curvature) outer rim.
Read the live K min / K max range for the current surface and mesh.

Did You Know?

Carl Friedrich Gauss called his 1827 discovery that curvature is intrinsic the Theorema Egregium — Latin for "remarkable theorem" — because it was genuinely surprising that a purely local, distance-based quantity could be so fundamental. It explains why every flat map of the round Earth must distort something (area, angle or distance), since a sphere's surface cannot be mapped to a plane without changing intrinsic measurements. The same idea explains why a slice of pizza held flat flops downward: folding it slightly forces a nonzero curvature along the fold, which, to keep the total Gaussian curvature consistent, flattens out (stiffens) the curvature running the length of the slice, holding it rigid.

About this simulation

This simulator renders five classic surfaces as rotatable, colour-coded 3D meshes. At every mesh vertex it evaluates the real Gaussian curvature formula for that surface's parametrization — not an approximation — and maps the value to a blue-white-red colour scale.

🔬 What it shows

Each surface is sampled on a (u, v) parameter grid; curvature is computed analytically (constant 1/R² for the sphere, the graph-curvature formula for the saddle, zero identically for the plane and cylinder, and cos(v) / (r(R + r·cos v)) for the torus) rather than estimated numerically.

🎮 How to use

Pick a surface, drag to rotate or enable auto-rotate, and raise Resolution for a finer mesh. Switch to the torus and move the r/R ratio slider to watch the positive (outer) and negative (inner) curvature bands shift.

💡 Did you know?

A cylinder has zero Gaussian curvature everywhere, exactly like a flat plane, which is precisely why you can roll a sheet of paper into a tube without stretching or tearing it.

Frequently asked questions

Why does a cylinder have zero Gaussian curvature?

Gaussian curvature is the product of the two principal curvatures. A cylinder curves around its circular cross-section but is perfectly straight along its length, so one principal curvature is zero. Multiplying by zero gives K = 0 everywhere, even though the surface is visibly bent in 3D space.

What does "intrinsic" curvature mean?

Intrinsic properties can be measured entirely within the surface, using only distances and angles along it, without reference to how the surface sits in 3D space. Gauss proved Gaussian curvature is intrinsic, so a flat plane and a cylinder are intrinsically identical (both K = 0), even though a cylinder looks curved from the outside.

Why does the torus have both positive and negative curvature?

On the outer rim of a torus, the surface bulges outward in both directions like a sphere, giving positive K. On the inner rim, it curves like a saddle, bulging one way and pinching the other, giving negative K. The formula K = cos(v) / (r(R + r·cos v)) captures this: cos(v) is positive on the outer rim and negative on the inner rim.

Why can't a flat map of Earth be perfectly accurate?

A sphere has constant positive Gaussian curvature, while a flat map has zero curvature. Since Gaussian curvature is intrinsic and preserved only by distance-preserving transformations, no flat map can represent a sphere without distorting at least one of area, angle or distance.

What is the Theorema Egregium?

It is Gauss's 1827 result proving that Gaussian curvature depends only on measurements taken within the surface itself, not on how it is embedded in 3D space. The name means "remarkable theorem" in Latin, reflecting how unexpected this result was at the time.

⚙ Under the hood

Compare a sphere, plane, saddle, cylinder and torus rendered as rotatable shaded meshes, colour-coded blue-to-red by real Gaussian curvature computed at every point from each surface's differential-geometry formula, including why a curved cylinder has zero curvature.

Canvas 2DGaussian CurvatureDifferential GeometryTorusTheorema Egregium

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