projection of right angle

Theorem.  The projection (http://planetmath.org/ProjectionOfPoint) of a right angle in ${ℝ}^3$ 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.

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