Doubly ruled surface

This article defines a property that makes sense for a surface embedded in

${\mathbb{R}}^3$

, viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.
View other such properties

Definition

A surface embedded in ${\mathbb{R}}^3$ is said to be doubly ruled if given any point in the surface, there are two distinct lines through the point, that lie completely on the surface.

Classification

Further information: Classification of doubly ruled surfaces

Doubly ruled surface	Equational/implicit description
Euclidean plane	$z=0$ (the $xy$-plane)
circular hyperboloid of one sheet	$\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{c^2}=1$
hyperbolic paraboloid	$z=\frac{y^2}{b^2}-\frac{x^2}{a^2}$

Relation with other properties

Weaker properties

• Ruled surface