Ruled Surfaces

A ruled surface is defined by the property that through every point in the surface, there is at least one straight line which also lies in the surface. A ruled surface may be thought of as one "swept out" by a straight line moving in space. To describe how such a line moves, first recall that any line is uniquely determined by two distinct points which lie on it. Then by choosing two curves, and a suitable map between their points, we can join up points with lines in order to define a ruled surface. Here is a nice illustration of this process [1]. 

Alternatively, we can imagine specifying a simple curve, called the "director curve", and a direction at each point on the curve. These directions are determined by the "director line" (a line in $\mathbb{R}^3$) by choosing a map $\mathbf{f}$ from points on the director curve to points on the director line. For each point on the director curve there is a unique line through $\mathbf{x}$ and $\mathbf{f}(\mathbf{x})$, and these lines can now be used to define a surface.

Algebraically speaking, we can label (parameterise) the points on the director curve by a real variable $t$, so points on it have the form $(x_{\alpha}(t), y_{\alpha}(t), z_{\alpha}(t))$. We can also parameterise with the variable $t$ the directions to the director line as (unit) vectors $(n_{1}(t), n_{2}(t), (n_{3}(t))$. Then any point on the surface can be written as: $$(x(s,t), y(s,t), z(s,t)) = (x_{\alpha}(t), y_{\alpha}(t), z_{\alpha}(t)) + s  (n_{1}(t), n_{2}(t), (n_{3}(t)).$$ This says that you set the point on the surface by starting from a point, labelled by $t$, on the director curve and travelling a distance $s$ along the director line through that point. Parameterisations like this are a natural and useful way to describe any surface; a familiar example is the use of latitude and longitude to specify locations on a sphere.

Cylinders and cones are simple examples of ruled surfaces. More complicated ruled surfaces are of interest to architects, especially with free-form architecture and complicated shapes [2]. There are numerous examples of ruled surface structures in contemporary architecture, including cooling towers (hyperboloid) and saddle roofs (hyperbolic paraboloid). String models of ruled surfaces have inspired many artworks, some of which were exhibited in a Royal Society exhibition [3]. See here for some interesting string sculptures, here for an artwork of a ruled surface in a cylinder, and here for a similarly designed rope art installation.

References

[1] http://www.mathcurve.com/surfaces/cubic/cubique_reglee.shtml

[2] Flöry et al, Ruled Surfaces for Rationalization and Design in Architecture (2010), http://www.geometrie.tuwien.ac.at/floery/papers/acadia_ruled.pdf

[3] The Royal Society, Intersections Henry Moore and stringed surfaces (2012), http://www.newton.ac.uk/files/attachments/1063471/179161.pdf