# ruled surface

A straight line $g$ moving continuously in space sweeps a 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

$$\overrightarrow{r}=\overrightarrow{a}(t),\overrightarrow{r}=\overrightarrow{b}(t).$$ |

Using two parameters $s$ and $t$ we express the position vector (http://planetmath.org/PositionVector) of an arbitrary point of the ruled surface as

$$\overrightarrow{r}=\overrightarrow{a}(t)+s\overrightarrow{b}(t).$$ |

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

Examples

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

$$\overrightarrow{r}=ct\overrightarrow{k}+s(\overrightarrow{i}\mathrm{cos}t+\overrightarrow{j}\mathrm{sin}t)=\left(\begin{array}{c}\hfill s\mathrm{cos}t\hfill \\ \hfill s\mathrm{sin}t\hfill \\ \hfill ct\hfill \end{array}\right).$$ |

2. The equation

$$z=xy$$ |

presents a hyperbolic paraboloid^{} (if we rotate the coordinate system^{} (http://planetmath.org/RotationMatrix) 45 about the $z$-axis using the formulae $x=({x}^{\prime}-{y}^{\prime})/\sqrt{2}$,
$y=({x}^{\prime}+{y}^{\prime})/\sqrt{2}$, the equation gets the form ${x}^{\prime 2}-{y}^{\prime 2}=2z$). Since the position vector of any point of the surface may be written using the parameters $s$ and $t$ as

$$\overrightarrow{r}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill t\hfill \\ \hfill 0\hfill \end{array}\right)+s\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill t\hfill \end{array}\right),$$ |

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 (http://planetmath.org/QuadraticSurfaces).

Title | ruled surface |

Canonical name | RuledSurface |

Date of creation | 2016-03-03 17:28:55 |

Last modified on | 2016-03-03 17:28:55 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 19 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 51M20 |

Classification | msc 51M04 |

Related topic | EquationOfPlane |

Related topic | GraphOfEquationXyConstant |

Defines | directrix |

Defines | base curve |

Defines | director curve |

Defines | generatrix |

Defines | generatrices |

Defines | ruling |

Defines | helicoid |