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[parent] projection of point (Definition)

Let a straight line $ \ell$ be given in a Euclidean plane or space. The projection of a point $ P$ onto the line $ \ell$ is the point $ P'$ of $ \ell$ at which the normal line of $ \ell$ passing through $ P$ intersects $ \ell$. One says that $ P$ has been projected onto the line $ \ell$.


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

The projection of a set $ S$ of points onto the line $ \ell$ is defined to be the set of projection points of all points of $ S$ on $ \ell$.

Especially, the projection of a line segment $ \overline{PQ}$ onto $ \ell$ is the line segment $ \overline{P'Q'}$ determined by the projection points $ P'$ and $ Q'$ of $ P$ and $ Q$. If the length of $ \overline{PQ}$ is $ a$ and the angle between the lines $ \overleftrightarrow{PQ}$ and $ \ell$ 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)... ...](-4.2,0){$Q$} \rput[l](-1.9,-2.1){$Q'$} \rput[r](2.8,3){$\ell$} \end{pspicture}



"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:  projection point
Also defines:  project, projection of line segment
Keywords:  orthogonal projection

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Cross-references: length, line segment, intersects, passing through, normal line, point, Euclidean plane, line
There are 37 references to this entry.

This is version 15 of projection of point, born on 2007-05-27, modified 2007-08-15.
Object id is 9475, canonical name is ProjectionOfPoint.
Accessed 3277 times total.

Classification:
AMS MSC51N99 (Geometry :: Analytic and descriptive geometry :: Miscellaneous)

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