ruled surface RuledSurface 2005-08-28 12:46:55 2007-10-22 17:41:46 Topic surface directrix base curve director curve generatrix generatrices ruling helicoid surface ruled % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} A straight line $g$ moving continuously in space sweeps a {\em ruled surface}.\, Formally:\, A surface $S$ in $\mathbb{R}^3$ is a ruled surface if it is connected and if for any point $p$ of $S$, there is a line $g$ such that\, $p\in g\subset S$. Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms $$\vec{r} = \vec{a}(t), \quad \vec{r} = \vec{b}(t).$$ Using two parameters $s$ and $t$ we express the position vector of an arbitrary point of the ruled surface as $$\vec{r} = \vec{a}(t)+ s\,\vec{b}(t).$$ Here\, $\vec{r} = \vec{a}(t)$\, is a curve on the ruled surface and is called \PMlinkescapetext{{\em directrix}} or the \PMlinkescapetext{{\em base curve}} of the surface, while\, $\vec{r} = \vec{b}(t)$\, is the {\em director curve} of the surface.\, Every position of $g$ is a {\em generatrix} or {\em ruling} of the ruled surface. \textbf{Examples} 1.\, Choosing the $z$-axis ($\vec{r} = ct\vec{k}$,\, $c \neq 0$) as the \PMlinkescapetext{directrix} and the unit circle ($\vec{r} = \vec{i}\cos{t}+\vec{j}\sin{t}$) as the director curve we get the {\em helicoid} (``screw surface''; cf. the circular helix) $$\vec{r} = ct\vec{k}+ s\,(\vec{i}\cos{t}+\vec{j}\sin{t}) = \left(\!\begin{array}{c}s\cos{t}\\ s\sin{t}\\ ct\end{array}\!\right)\!.$$ \begin{figure}[!htb] \begin{center} \includegraphics{helicoid.eps} \end{center} \caption{The helicoid as a ruled surface} \end{figure} 2.\, The equation\, $$z = xy$$ presents a hyperbolic paraboloid (if we \PMlinkname{rotate the coordinate system}{RotationMatrix} 45 \PMlinkescapetext{degrees} about the $z$-axis using the formulae\, $x = (x'-y')/\sqrt{2}$,\, $y = (x'+y')/\sqrt{2}$,\, the equation gets the form\, $x'^2-y'^2 = 2z$).\, Since the position vector of any point of the surface may be written using the parameters $s$ and $t$ as $$\vec{r} = \left(\!\begin{array}{c}0\\ t\\ 0\end{array}\!\right)\! +s\left(\!\begin{array}{c}1\\ 0\\ t\end{array}\!\right)\!,$$ we see that it's a question of a ruled surface with rectilinear directrix and director curve. 3.\, Other ruled surfaces are for example all cylindrical surfaces (plane included), conical surfaces, one-sheeted hyperboloid.