R one pavilion

R ONE

PAVILION

RULED SURFACE PAVILION A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle.

A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points.

A hyperbolic paraboloid roof of Warszawa Ochota railway station in Warsaw, Poland.

A hyperboloid Kobe Port Tower, Kobe, Japan, with a double ruling.

Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements

The gridshell of Shukhov Tower in Moscow, whose sections are doubly ruled.

The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are also sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case “straight line� is understood to mean an affine or projective line.