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Viewing Version
8
of
'envelope'
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| Title of object: |
envelope |
| Canonical Name: |
Envelope |
| Type: |
Definition |
| Created on: |
2007-05-29 14:19:00 |
| Modified on: |
2007-05-30 05:34:17 |
| Classification: |
msc:26A24, msc:26B05, msc:51N20 |
| Keywords: |
family of curves |
Revision comment (for changes between this and next version):
| some more adjustments in picture |
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:
Two plane curves are said to {\em touch each other} or {\em have a tangency} at a point if they have a common tangent at that point.
The {\em envelope} of a family of plane curves is a curve which touches in each of its points one of the curves of the family.
For example, the envelope of the family\, $y = mx-\sqrt{1+m^2}$,\, with $m$ the parameter, is the \PMlinkname{open}{OpenSet} lower semicircle of the unit circle.\, Indeed, the distance of any line
$$mx-y-\sqrt{1+m^2} = 0$$
of the family from the center of the unit circle is
$$\frac{|m\cdot0-1\cdot0-\sqrt{1+m^2}|}{\sqrt{m^2+(-1)^2}} = 1,$$
whence the line is the tangent to the circle.
Below are some examples. The red curve is the lower semicircle of the unit circle, the black lines belong to the family\, $y=mx-\sqrt{1+m^2}$,\, and the equation of each line is given.
\begin{center}
\begin{pspicture}(-3,-3)(3,0)
\psarc[linecolor=red]{o-o}(0,0){2}{180}{360}
\psline{-}(-0.1716,-3)(3,0.1716)
\rput[l](2.9,-0.1716){$y=x+\sqrt{2}$}
\psline{-}(-3,-2)(3,-2)
\rput[a](-3,-2.3){$y=-1$}
\psline{-}(-2.3094,0)(-0.577,-3)
\rput[r](-2.3094,0){$y=-\sqrt{3}x-2$}
\rput[b](-0.577,-3){.}
\end{pspicture}
\end{center} |
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