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'projection of point'
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| Title of object: |
projection of point |
| Canonical Name: |
ProjectionOfPoint |
| Type: |
Definition |
| Created on: |
2007-05-27 11:29:54 |
| Modified on: |
2007-06-13 01:59:34 |
| Classification: |
msc:51N99 |
| Keywords: |
orthogonal projection |
| Defines: |
project, projection of line segment |
| Synonyms: |
projection of point=projection point |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\PMlinkescapeword{projection}
Let a \PMlinkescapetext{straight} line $\ell$ be given in a Euclidean plane or space. The {\em projection of a \PMlinkescapetext{point}} $P$ on 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 {\em projected} on the line $\ell$.
\begin{center}
\begin{pspicture}(-3,-3)(3,3)
\rput[b](-3,-3){.}
\rput[a](3,3){.}
\psline(-3,-3)(3,3)
\psline[linestyle=dashed](-2,2)(0,0)
\psline(-0.3,0.3)(0,0.6)
\psline(0,0.6)(0.3,0.3)
\psdots(-2,2)(0,0)
\rput[r](-2.2,2){$P$}
\rput[l](0.1,-0.1){$P'$}
\rput[r](2.8,3){$\ell$}
\end{pspicture}
\end{center}
The {\em projection of a set} $S$ of points on the line $\ell$ 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}} $\overline{PQ}$ on $\ell$ is the line segment $\overline{P'Q'}$ 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 $\ell$ is $\alpha$, then the length $p$ of its projection is
$$p\, =\, a\,\cos\alpha.$$
\begin{center}
\begin{pspicture}(-7,-7)(3,3)
\rput[b](-7,-7){.}
\rput[a](3,3){.}
\psline(-7,-7)(3,3)
\psline[linestyle=dashed](-3,3)(0,0)
\psline(-0.3,0.3)(0,0.6)
\psline(0,0.6)(0.3,0.3)
\psline[linestyle=dashed](-4,0)(-2,-2)
\psline(-2.3,-1.7)(-2,-1.4)
\psline(-2,-1.4)(-1.7,-1.7)
\psline[linecolor=red](-3,3)(-4,0)
\psline[linecolor=red,linestyle=dashed](-4,0)(-6,-6)
\psarc(-6,-6){0.5}{45}{71.565}
\rput[b](-5.5,-5.4){$\alpha$}
\psline[linecolor=blue](0,0)(-2,-2)
\psdots(-3,3)(0,0)(-4,0)(-2,-2)
\rput[r](-3.2,3){$P$}
\rput[l](0.1,-0.1){$P'$}
\rput[r](-4.2,0){$Q$}
\rput[l](-1.9,-2.1){$Q'$}
\rput[r](2.8,3){$\ell$}
\end{pspicture}
\end{center} |
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