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Revision difference : envelope |
| Version 15 |
Version 14 |
| 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.\\ |
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, 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, 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} |
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\rput(3,0.1716){.} |
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\rput(-0.1716,-3){.} |
| \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(-2.3094,0){.} |
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\rput(-0.577,-3){.} |
| \rput[r](-2.3094,-0.1716){$y=-x\sqrt{3}-2$} |
\rput[r](-2.3094,-0.1716){$y=-x\sqrt{3}-2$} |
| \rput[b](-0.577,-3){.} |
\rput[b](-0.577,-3){.} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| Let $c$ be the parameter of the family\, $F(x,\,y,\,c) = 0$\, of curves and suppose that the function $F$ has the partial derivatives $F'_x$, $F'_y$ and $F'_c$ in a certain domain of $\mathbb{R}^3$.\, If the family has an envelope $E$ in this domain, then the coordinates $x,\,y$ of an arbitrary point of $E$ and the value $c$ of the parameter determining the family member touching $E$ in\, $(x,\,y)$\, satisfy the pair of equations |
Let $c$ be the parameter of the family\, $F(x,\,y,\,c) = 0$\, of curves and suppose that the function $F$ has the partial derivatives $F'_x$, $F'_y$ and $F'_c$ in a certain domain of $\mathbb{R}^3$.\, If the family has an envelope $E$ in this domain, then the coordinates $x,\,y$ of an arbitrary point of $E$ and the value $c$ of the parameter determining the family member touching $E$ in\, $(x,\,y)$\, satisfy the pair of equations |
| \begin{align*} |
\begin{align*} |
| \begin{cases} |
\begin{cases} |
| F(x,\,y,\,c) = 0,\\ |
F(x,\,y,\,c) = 0,\\ |
| F'_c(x,\,y,\,c) = 0. |
F'_c(x,\,y,\,c) = 0. |
| \end{cases} |
\end{cases} |
| \end{align*} |
\end{align*} |
| I.e., one may in principle eliminate $c$ from such a pair of equations and obtain the equation of an envelope. |
I.e., one may in principle eliminate $c$ from such a pair of equations and obtain the equation of an envelope. |
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| \textbf{Example.} |
\textbf{Example.} |
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