Developable surface

Builds on Regular surface Gaussian curvature

Definition

Let $S \subset \R^3$ be a regular surface with Gaussian curvature $K \colon S \to \R$ . The surface $S$ is developable if $K(p) = 0$ for every point $p \in S$ .

Remarks

The cylinder $x^2 + y^2 = 1$ is developable. Its principal curvatures are $1$ along the horizontal circles and $0$ along the vertical lines, so $K = 1 \cdot 0 = 0$ . The sphere of radius $r$ is not developable, since $K = 1/r^2 \neq 0$ at every point.

By the Theorema Egregium, $K$ depends only on the first fundamental form. A surface with $K = 0$ is locally isometric to the plane, and conversely. A developable surface without planar points is a ruled surface. Away from a small exceptional set, it is locally a piece of a cylinder, a cone or a tangent developable. Some authors define a developable surface as a ruled surface with $K = 0$ .

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Tag	`PA4HJDEP28`
URI	`m2://PA4HJDEP28`
Kind	definition
Writing	W0 stub
MSC	53A05
Defines	Developable surface
Uses	Regular surface, Gaussian curvature
Changed	2 November 2026

Developable surface

Definition

Remarks

Reading path

Further reading

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