ruled surface

String model of a hyperboloid

String model of a hyperboloid.

A ruled surface is a surface that is built up from an infinite number of perfectly straight lines. A cylinder, for example, is a ruled surface of parallel straight lines. A cone is a ruled surface of straight lines that meet at the apex of the cone.


Also known as scrolls, ruled surfaces have been studied for centuries by geometers such as the Jesuits Roger Boscovich and Andre Tacquet as well as by their famous students, including Gaspar Monge and Phillippe de Lahire. Examples that stand out because of both their striking shape and their relative ease of construction include hyperboloids, helicoids and Möbius bands. Most ruled surfaces, however, are so complicated that, before the computer age, they were almost impossible to construct.