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 $AP^{\prime}B$ would be a right angle, the point $P^{\prime}$ must be on a circle of the plane $\pi$ having $AB$ as diameter.  But $OP^{\prime}$ is as the projection of the segment $OP$ shorter than $OP$.  It follows that the angle $AP^{\prime}B$ 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 ProjectionOfRightAngle 2013-03-22 19:20:51 2013-03-22 19:20:51 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 51N99 msc 51N20 AngleBetweenLineAndPlane AngleOfView AngleOfViewOfALineSegment