# projection of point

Let a line $l$ be given in a Euclidean plane^{} or space. The (orthogonal^{}) projection of a $P$ onto the line $l$ is the point ${P}^{\prime}$ of $l$ at which the normal line of $l$ passing through $P$ intersects $l$. One says that $P$ has been (orthogonally) projected onto the line $l$.

The projection of a set $S$ of points onto the line $l$ is defined to be the set of projection points of all points of $S$ on $l$.

Especially, the projection of a $PQ$ onto $l$ is the line segment^{} ${P}^{\prime}{Q}^{\prime}$ determined by the projection points ${P}^{\prime}$ and ${Q}^{\prime}$ of $P$ and $Q$. If the length of $PQ$ is $a$ and the angle between the lines (http://planetmath.org/AngleBetweenTwoLines) $PQ$ and $l$ is $\alpha $, then the length $p$ of its projection is

$p=a\mathrm{cos}\alpha .$ | (1) |

Remark. As one speaks of the projections onto a line $l$, one can speak in the Euclidean space also of projections onto a plane $\tau $.

Title | projection of point |

Canonical name | ProjectionOfPoint |

Date of creation | 2013-03-22 17:09:50 |

Last modified on | 2013-03-22 17:09:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 21 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N99 |

Synonym | orthogonal projection |

Related topic | Projection |

Related topic | CompassAndStraightedgeConstructionOfPerpendicular |

Related topic | MeusniersTheorem |

Defines | project |

Defines | projection of line segment |