projection of point

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

$p=a\cos \alpha .$ (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