---
tag: PA4HJDEP28
kind: definition
name: Developable surface
msc: [53A05]
defines: YKVHDB52XW
uses: [6080X3FAD4, SQVQ5DQT6W]
sources:
  - url: https://en.wikipedia.org/wiki/Developable_surface
---

## Definition

Let $S \subset \R^3$ be a [[regular surface|6080X3FAD4]] with [[Gaussian curvature|SQVQ5DQT6W]] $K \colon S \to \R$.
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|YKSA8SJ81V]], $K$ 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 $K = 0$.

## Explanations

### L1 human

A developable surface is one with zero Gaussian curvature everywhere.
At each point one of the two principal curvatures is zero, so the surface bends in only one direction at a time.