# projection of right angle

Theorem. The projection (http://planetmath.org/ProjectionOfPoint) of a right angle^{} in ${\mathbb{R}}^{3}$ onto a plane is a right angle if and only if at least one of its sides is parallel^{} to the plane.

*Proof.* Consider the projection of an angle $\alpha $ with vertex (http://planetmath.org/Angle) $P$ onto the plane $\pi $. Let ${P}^{\prime}$ be the projection of $P$ onto $\pi $. If neither of the sides of $\alpha $ is parallel to $\pi $, then the lines of the sides intersect the plane in two distinct points $A$ and $B$. In order to that the angle of view of the segment $AB$ seen from the point $P$ would be a right angle, $P$ must be on a sphere with diameter^{} $AB$ centered at a point $O$. In order to that the projection angle $A{P}^{\prime}B$ would be a right angle, the point ${P}^{\prime}$ must be on a circle of the plane $\pi $ having $AB$ as diameter. But $O{P}^{\prime}$ is as the projection of the segment $OP$ shorter than $OP$. It follows that the angle $A{P}^{\prime}B$ is obtuse and hence cannot be right.

On the other hand, it’s not hard to see that the projection of a right angle is a right angle always when one or both of its sides are parallel to the projection plane.

## References

- 1 E. J. Nyström: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).

Title | projection of right angle |
---|---|

Canonical name | ProjectionOfRightAngle |

Date of creation | 2013-03-22 19:20:51 |

Last modified on | 2013-03-22 19:20:51 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 51N99 |

Classification | msc 51N20 |

Related topic | AngleBetweenLineAndPlane |

Related topic | AngleOfView |

Related topic | AngleOfViewOfALineSegment |