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'normal line'
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| Title of object: |
normal line |
| Canonical Name: |
NormalLine |
| Type: |
Definition |
| Created on: |
2007-05-27 12:46:33 |
| Modified on: |
2007-05-27 14:25:42 |
| Classification: |
msc:26A24, msc:26B05 |
| Synonyms: |
normal line=normal |
Revision comment (for changes between this and next version):
| bottom dot is not cooperating.... |
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:
A {\em normal line} of a curve at one of its points $P$ is the line through this point and perpendicular to the tangent line of the curve at $P$.
If the plane curve\, $y = f(x)$\, has a non-horizontal 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
$$y-f(x_0) = -\frac{1}{f'(x_0)}(x-x_0).$$
In the case that the tangent is horizontal, the equation of the vertical normal is
$$x = x_0.$$
The normal of a curve at its point $P$ goes always through the center of curvature belonging to 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 $P$.
\begin{center}
\begin{pspicture}(-2,-1)(2,4)
\parabola{<->}(2,4)(0,0)
\rput[b](-2,4){.}
\rput[b](2,4){.}
\rput[l](-0.1,-0.9){.}
\psline[linecolor=red]{<->}(0,-1)(2,3)
\psline[linecolor=blue]{<->}(-2,2.5)(2,0.5)
\psdot(1,1)
\rput[b](0.9,1.2){$P$}
\end{pspicture}
\end{center} |
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