projection of right angle

Theorem.  The projection ( of a right angleMathworldPlanetmathPlanetmath in 3 onto a plane is a right angle if and only if at least one of its sides is parallelMathworldPlanetmathPlanetmath to the plane.

Proof.  Consider the projection of an angle α with vertex ( P onto the plane π.  Let P be the projection of P onto π.  If neither of the sides of α is parallel to π, 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 diameterMathworldPlanetmathPlanetmath AB centered at a point O.  In order to that the projection angle APB would be a right angle, the point P must be on a circle of the plane π having AB as diameter.  But OP is as the projection of the segment OP shorter than OP.  It follows that the angle APB 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.


  • 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