projection of pointProjectionOfPoint2007-05-27 11:29:542009-11-17 15:33:00Definitionprojectionprojectprojection of line segmentorthogonal projection% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{onto} \PMlinkescapeword{projection}
Let a \PMlinkescapetext{straight} line $l$ be given in a Euclidean plane or space. The ({\em orthogonal}) {\em projection of a \PMlinkescapetext{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 ({\em orthogonally}) {\em projected} onto the line $l$.
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The {\em 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 {\em projection of a \PMlinkescapetext{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 \PMlinkname{angle between the lines}{AngleBetweenTwoLines} $PQ$ and $l$ is $\alpha$, then the length $p$ of its projection is
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p \;=\; a\,\cos\alpha.
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\rput[l](-1.9,-2.1){$Q'$}
\rput[r](2.8,3){$l$}
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\textbf{Remark.}\, As one speaks of the projections onto a line $l$, one can speak in the Euclidean space also of {\em projections onto a plane} $\tau$.