2. Ruled surfaces

A ruled surface is a surface that is swept out by a moving straight line, for example, the ribbon and the Möbius strip. One simple way to get a ruled surface is to take two parametric curves in $\mathbb{E}^ 3$, say A$(t)$ and B$(t)$, for the same $t$-interval $[a,b]$, and at each time $t$ draw a line between them using a new parameter $u$, to make a ``linear blending'' of A$(t)$ and B$(t)$.

P$(t,u) = (1-u)$A$(t) + u$B$(t)$, for $0 \leq u \leq 1$ and $a \leq t \leq b$.

Example 8. A hyperboloid of one sheet: Let A$(t)$ be a circle in the $z = -1$ plane, and let B$(t)$ be a circle in the $z = 1$ plane, traced out $90 ^\circ$ out of phase with the first one. Thus

A$(t) = \left[\begin{array}{c}\cos t\\ \sin t\\ -1\end{array}\right]$ and

B$(t) = \left[\begin{array}{c}\cos(t + \frac {\pi}2)\\ \sin(t + \frac {\pi}2)\\ 1\end{array}\right]$, for $0 \leq t \leq 2 \pi$.

Example 9. A self-intersecting surface: Let A$(t)$ be a circle in the $z = -1$ plane, and let B$(t)$ be a circle in the $z = 1$ plane, swept out twice as the first circle is swept out once. Thus A$(t) = \left[\begin{array}{c}\cos t\\ \sin t\\ -1\end{array}\right]$ and B$(t) = \left[\begin{array}{c}\cos 2t\\ \sin 2t\\ 1\end{array}\right]$, for $0 \leq t \leq 2 \pi$. Horizontal cross sections are limaçons; one is a cardioid.

Example 10. A bilinear patch: Let $P _ {00}, P _ {01}, P _ {10}, P _ {11}$ be four points in E$^ 3$. What is the simplest patch parameterized by a rectangle that has these points as corners? Just let A$(t)$ be the line segment from $P _ {00}$ to $P _ {10}$ and let B$(t)$ be the line segment from $P _ {01}$ to $P _ {11}$. After simplifying, you get

P$(t,u) = (1-t)(1-u)P _ {00} + t(1-u)P _ {10} + (1-t)uP _ {01} + tu P _ {11}$.

This can be plotted for $0 \leq t \leq 1$ and $0 \leq u \leq 1$. You can check that this works. This function is quadratic, in that there are terms involving $tu$, but it is bilinear in the sense that if either parameter is held fixed the function in the other parameter is linear. P$(t,u)$ is called a bilinear patch. A neater way to write it is to use matrices, allowing points (pairs of numbers) as entries:

P$(t,u) = \left[\begin{array}{cc}(1-t)&t\end{array}\right]\left[\begin{array}{cc... ...& P _ {11}\end{array}\right] \left[\begin{array}{c}(1-u)\\ u\end{array}\right]$

Even though the surface is curved, it contains two natural families of straight lines, namely the lines obtained by holding one of $t$ and $u$ constant and varying the other (``isoparametric curves'', as in §6 below). Therefore the surface is ruled.