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Viewing Version
5
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'angle of view of a line segment'
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| Title of object: |
angle of view of a line segment |
| Canonical Name: |
AngleOfViewOfALineSegment |
| Type: |
Topic |
| Created on: |
2007-10-04 11:28:37 |
| Modified on: |
2007-10-10 13:55:27 |
| Classification: |
msc:51G05, msc:51F20 |
| Defines: |
angle of view |
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}
% 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
\usepackage{pstricks}
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
Let $PQ$ be a line segment and $A$ a point not belonging to $PQ$. Let the magnitude of the angle $PAQ$ be $\alpha$. One says that the line segment $PQ$ {\em is seen from the point $A$ in an angle of $\alpha$}; one may also speak of the {\em angle of view} of $PQ$.
The locus of the points from which a given line segment $PQ$ is seen in an angle of $\alpha$ (with\, $0 < \alpha < 180^\circ$) consists of two congruent circular arcs having the line segment as the common chord and containing the circumferential angles equal to $\alpha$.
Especially, the locus of the points from which the line segment is seen in an angle of $90^\circ$ is the circle having the line segment as its diameter.
\begin{center}
\begin{pspicture}(-3,-3)(3,3)
\psline[linecolor=blue](-1.73,0)(1.73,0)
\rput[a](-2.1,-0.1){$P$}
\rput[a](2.1,-0.1){$Q$}
\psarc[linecolor=red](0,1){2}{-30}{210}
\psarc[linecolor=red](0,-1){2}{-210}{30}
\psline(-1.73,0)(-1.2,2.6)
\psline(1.73,0)(-1.2,2.6)
\psline(-1.73,0)(2,1)
\psline(1.73,0)(2,1)
\rput[a](-1.08,2.25){$\alpha$}
\rput[a](1.72,0.75){$\alpha$}
\psdots[linecolor=blue](-1.73,0)(1.73,0)
\psdots[linecolor=red](-1.2,2.6)(2,1)
\end{pspicture}
\end{center} |
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