| Version 10 |
Version 9 |
| 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. |
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. |
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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.\\
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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. |
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| 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 |
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 |
| $$mx-y-\sqrt{1+m^2} = 0$$ |
$$mx-y-\sqrt{1+m^2} = 0$$ |
| of the family from the center of the unit circle is |
of the family from the center of the unit circle is |
| $$\frac{|m\cdot0-1\cdot0-\sqrt{1+m^2}|}{\sqrt{m^2+(-1)^2}} = 1,$$ |
$$\frac{|m\cdot0-1\cdot0-\sqrt{1+m^2}|}{\sqrt{m^2+(-1)^2}} = 1,$$ |
| whence the line is the tangent to the circle. |
whence the line is the tangent to the circle. |
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| 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. |
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. |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-3,-3)(3,0) |
\begin{pspicture}(-3,-3)(3,0) |
| \psarc[linecolor=red]{o-o}(0,0){2}{180}{360} |
\psarc[linecolor=red]{o-o}(0,0){2}{180}{360} |
| \psline{-}(-0.1716,-3)(3,0.1716) |
\psline{-}(-0.1716,-3)(3,0.1716) |
| \rput[l](2.9,-0.1716){$y=x+\sqrt{2}$} |
\rput[l](2.9,-0.1716){$y=x+\sqrt{2}$} |
| \psline{-}(-3,-2)(3,-2) |
\psline{-}(-3,-2)(3,-2) |
| \rput[a](-2.3,-2.3){$y=-1$} |
\rput[a](-2.3,-2.3){$y=-1$} |
| \psline{-}(-2.3094,0)(-0.577,-3) |
\psline{-}(-2.3094,0)(-0.577,-3) |
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\rput[r](-2.3094,-0.1716){$y=-x\sqrt{3}-2$}
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\rput[r](-2.3094,-0.1716){$y=-\sqrt{3}x-2$}
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| \rput[b](-0.577,-3){.} |
\rput[b](-0.577,-3){.} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
