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Curve Point Incidence Geometry Equation Parameter Circle Convex Angle Connected Space Surface Cone In Mathbb R Cylinder Rotation Matrix Helix Position Vector Quadratic Surfaces

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Ruled Surface

A straight line moving continuously in space sweeps a ruled surface. Formally: A surface in $\mathbb{R}^3$ is a ruled surface if it is connected and if for any point of , there is a line 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

$\displaystyle \vec{r} = \vec{a}(t), \quad \vec{r} = \vec{b}(t).$

Using two parameters

and

we express the position vector of an arbitrary point of the ruled surface as

$\displaystyle \vec{r} = \vec{a}(t)+ s \vec{b}(t).$

Here $\vec{r} = \vec{a}(t)$ is a curve on the ruled surface and is called directrix or the base curve of the surface, while $\vec{r} = \vec{b}(t)$ is the director curve of the surface. Every position of

is a generatrix or ruling of the ruled surface.

Examples

1. Choosing the -axis ( $\vec{r} = ct\vec{k}$ , $c \neq 0$ ) as the directrix and the unit circle ( $\vec{r} = \vec{i}\cos{t}+\vec{j}\sin{t}$ ) as the director curve we get the helicoid (“screw surface”; cf. the circular helix)

$\displaystyle \vec{r} = ct\vec{k}+ s (\vec{i}\cos{t}+\vec{j}\sin{t}) = \left(\!\begin{array}{c}s\cos{t}\ s\sin{t}\ ct\end{array}\!\right)\!.$

**Figure:** The helicoid as a ruled surface
$\includegraphics{helicoid.eps}$

2. The equation

$\displaystyle z = xy$

presents a hyperbolic paraboloid (if we rotate the coordinate system 45 degrees about the

-axis using the formulae $x = (x'-y')/\sqrt{2}$ , $y = (x'+y')/\sqrt{2}$ , the equation gets the form

). Since the position vector of any point of the surface may be written using the parameters

and

$\displaystyle \vec{r} = \left(\!\begin{array}{c}0\ t\ 0\end{array}\!\right)\! +s\left(\!\begin{array}{c}1\ 0\ t\end{array}\!\right)\!,$

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.

1	Circle
1	Curve
1	Incidence Geometry
1	Point
1	Convex Angle
1	Parameter
1	Equation
2	Surface
2	Connected Space
3	Cone In Mathbb R
4	Cylinder
6	Rotation Matrix
7	Position Vector
8	Quadratic Surfaces
13	Helix