Let a straight line $l$ be given in a Euclidean plane or space. The (orthogonal) projection of a point $P$ onto the line $l$ is the point $P'$ 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 line segment $PQ$ onto $l$ is the line segment $P'Q'$ determined by the projection points $P'$ and $Q'$ of $P$ and $Q$ . If the length of $PQ$ is $a$ and the angle between the lines $PQ$ and $l$ is $\alpha$ , then the length $p$ of its projection is

(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 $\tau$ .