Nondevelopable

Nondevelopable is a term used in geometry to describe a surface that cannot be flattened onto a plane without distortion. In contrast, a developable surface can be unrolled into a plane without stretching, tearing, or compressing. The distinction hinges on Gaussian curvature, an intrinsic measure of a surface’s curvature that depends only on distances along the surface.

A surface is developable if its Gaussian curvature is zero at every point; nondevelopable surfaces have nonzero Gaussian curvature on at least some region. This intrinsic curvature means that, when trying to lay the surface onto a plane, at least some distances or angles must change. Common examples of nondevelopable surfaces include spheres and ellipsoids, as well as many saddle-shaped surfaces such as the hyperbolic paraboloid; a torus is nondevelopable overall because its curvature varies in sign across the surface.

Gauss's Theorema Egregium establishes that Gaussian curvature is intrinsic, so flattening a nondevelopable surface would necessarily alter measurements like lengths and angles along the surface. This contrasts with developable surfaces, such as planes, cylinders, and cones, which can be bent from a plane without stretching.

Applications and implications: In cartography, the Earth’s curved surface cannot be represented on a flat map without some distortion, which is why map projections trade off different properties. In manufacturing and architecture, bending a sheet into a nondevelopable shape typically requires stretching or thinning, whereas developable surfaces can often be formed by simple bending of a flat sheet.