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# projection of point

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 $P^{{\prime}}$ 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 line segment $PQ$ 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 $PQ$ 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 $\tau$.

## Mathematics Subject Classification

51N99*no label found*

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