projection of point

Let a line $l$ be given in a Euclidean plane or space. The () 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

 $\displaystyle p\;=\;a\,\cos\alpha.$ (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