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[parent] projection of point (Definition)

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$.


\begin{pspicture}(-3,-3)(3,3) \rput[b](-3,-3){.} \rput[a](3,3){.} \psline(-3,-3)... ...ut[r](-2.2,2){$P$} \rput[l](0.1,-0.1){$P'$} \rput[r](2.8,3){$l$} \end{pspicture}

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

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

\begin{pspicture}(-7,-7)(3,3) \rput[b](-7,-7){.} \rput[a](3,3){.} \psline(-7,-7)... ...t[r](-4.2,0){$Q$} \rput[l](-1.9,-2.1){$Q'$} \rput[r](2.8,3){$l$} \end{pspicture}

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$.



"projection of point" is owned by pahio. [ full author list (3) ]
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See Also: projection, compass and straightedge construction of perpendicular, Meusnier's theorem

Other names:  orthogonal projection
Also defines:  project, projection of line segment
Keywords:  orthogonal projection

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projection formula (Theorem) by pahio
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Cross-references: plane, Euclidean space, projections, length, line segment, intersects, passing through, normal line, point, orthogonal, Euclidean plane, line
There are 38 references to this entry.

This is version 17 of projection of point, born on 2007-05-27, modified 2008-10-01.
Object id is 9475, canonical name is ProjectionOfPoint.
Accessed 5192 times total.

Classification:
AMS MSC: 51N99 (Geometry :: Analytic and descriptive geometry :: Miscellaneous)
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