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Revision difference : normal line |
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Version 10 |
| \PMlinkescapeword{word} |
\PMlinkescapeword{word} |
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| A {\em normal line} of a curve at one of its points $P$ is the line passing through this point and perpendicular to the tangent line of the curve at $P$. The word {\em normal} (without the word line immediately after it) is used to refer to a number line when it is clear from context that the word normal refers to a line. |
A {\em normal line} of a curve at one of its points $P$ is the line passing through this point and perpendicular to the tangent line of the curve at $P$. The word {\em normal} (without the word line immediately after it) is used to refer to a number line when it is clear from context that the word normal refers to a line. |
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| If the plane curve\, $y = f(x)$\, has a skew tangent at the point\, $(x_0,\,f(x_0))$,\, then the slope of the tangent at that point is\, $f'(x_0)$\, and the slope of the normal at that point is\, $\displaystyle -\frac{1}{f'(x_0)}$.\, The equation of the normal is thus |
If the plane curve\, $y = f(x)$\, has a skew tangent at the point\, $(x_0,\,f(x_0))$,\, then the slope of the tangent at that point is\, $f'(x_0)$\, and the slope of the normal at that point is\, $\displaystyle -\frac{1}{f'(x_0)}$.\, The equation of the normal is thus |
| $$y-f(x_0) = -\frac{1}{f'(x_0)}(x-x_0).$$ |
$$y-f(x_0) = -\frac{1}{f'(x_0)}(x-x_0).$$ |
| In the case that the tangent is horizontal, the equation of the vertical normal is |
In the case that the tangent is horizontal, the equation of the vertical normal is |
| $$x = x_0,$$ |
$$x = x_0,$$ |
| and in the case that the tangent is vertical, the equation of the normal is |
and in the case that the tangent is vertical, the equation of the normal is |
| $$y = f(x_0).$$ |
$$y = f(x_0).$$ |
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| The normal of a curve at its point $P$ always goes through the center of curvature belonging to the point $P$. |
The normal of a curve at its point $P$ always goes through the center of curvature belonging to the point $P$. |
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| In the picture below, the black curve is a parabola, the red line is the tangent at the point $P$, and the blue line is the normal at the point $P$. |
In the picture below, the black curve is a parabola, the red line is the tangent at the point $P$, and the blue line is the normal at the point $P$. |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-2,-1)(2,4) |
\begin{pspicture}(-2,-1)(2,4) |
| \parabola{-}(2,4)(0,0) |
\parabola{-}(2,4)(0,0) |
| \rput[b](-2,4){.} |
\rput[b](-2,4){.} |
| \rput[b](2,4){.} |
\rput[b](2,4){.} |
| \rput[l](-0.05,-1){.} |
\rput[l](-0.05,-1){.} |
| \psline[linecolor=red](0,-1)(2,3) |
\psline[linecolor=red](0,-1)(2,3) |
| \psline[linecolor=blue](-2,2.5)(2,0.5) |
\psline[linecolor=blue](-2,2.5)(2,0.5) |
| \psdot(1,1) |
\psdot(1,1) |
| \rput[b](0.9,1.2){$P$} |
\rput[b](0.9,1.2){$P$} |
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\psline[linewidth=0.2pt](0.9,0.8)(1.1,0.7)
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\psline(0.9,0.8)(1.1,0.7)
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\psline[linewidth=0.2pt](1.1,0.7)(1.2,0.9)
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\psline(1.1,0.7)(1.2,0.9)
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| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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