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