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| 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\, $p\in g\subseteq S$. |
A straight line $g$ moving continuously in space sweeps a {\em ruled surface}.\, Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms |
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| Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms |
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| $$\vec{r} = \vec{a}(t), \quad \vec{r} = \vec{b}(t).$$ |
$$\vec{r} = \vec{a}(t), \quad \vec{r} = \vec{b}(t).$$ |
| Using two parameters $s$ and $t$ we express the position vector of an arbitrary point of the ruled surface as |
Using two parameters $s$ and $t$ we express the position vector of an arbitrary point of the ruled surface as |
| $$\vec{r} = \vec{a}(t)+ s\,\vec{b}(t).$$ |
$$\vec{r} = \vec{a}(t)+ s\,\vec{b}(t).$$ |
| Here\, $\vec{r} = \vec{a}(t)$\, is a curve on the ruled surface and is called \PMlinkescapetext{{\em directrix}} or the |
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. |
\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. |
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| \textbf{Example.}\, Choosing the $z$-axis ($\vec{r} = t\vec{k}$) as the \PMlinkescapetext{directrix} and the unit circle ($\vec{r} = \vec{i}\cos{t}+\vec{j}\sin{t}$) |
\textbf{Example.}\, Choosing the $z$-axis ($\vec{r} = t\vec{k}$) 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 helix} or {\em helicoid} (``screw surface'') |
as the director curve we get the {\em helix} or {\em helicoid} (``screw surface'') |
| $$\vec{r} = t\vec{k}+ s\,(\vec{i}\cos{t}+\vec{j}\sin{t}) = |
$$\vec{r} = t\vec{k}+ s\,(\vec{i}\cos{t}+\vec{j}\sin{t}) = |
| \left(\!\begin{array}{c}s\cos{t}\\ s\sin{t}\\ t\end{array}\!\right)\!.$$ |
\left(\!\begin{array}{c}s\cos{t}\\ s\sin{t}\\ t\end{array}\!\right)\!.$$ |
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Other ruled surfaces are all cylindrical and conical surfaces, one-sheeted hyperboloid and hyperbolic paraboloid (note that a plane is a cylindrical surface).
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Other ruled surfaces are all cylindrical and conical surfaces, one-sheeted hyperboloid and hyperbolic paraboloid.
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