PlanetMath: normal line

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normal line

(Definition)

A normal line of a curve at one of its points is the line passing through this point and perpendicular to the tangent line of the curve at .

If the plane curve has a skew tangent at the point $(x_0,\,f(x_0))$ , then the slope of the tangent at that point is and the slope of the normal at that point is $\displaystyle -\frac{1}{f'(x_0)}$ . The equation of the normal is thus

$\displaystyle y-f(x_0) = -\frac{1}{f'(x_0)}(x-x_0).$

In the case that the tangent is horizontal, the equation of the vertical normal is

$\displaystyle x = x_0,$

and in the case that the tangent is vertical, the equation of the normal is

$\displaystyle y = f(x_0).$

The normal of a curve at its point always goes through the center of curvature belonging to the point .

In the picture below, the black curve is a parabola, the red line is the tangent at the point , and the blue line is the normal at the point .

$\begin{pspicture}(-2,-1)(2,4) \parabola{-}(2,4)(0,0) \rput[b](-2,4){.} \rput[b](... ...pt](0.9,0.8)(1.1,0.7) \psline[linewidth=0.2pt](1.1,0.7)(1.2,0.9) \end{pspicture}$

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"normal line" is owned by pahio. [ full author list (3) ]

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Other names:

normal of curve, normal

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Cross-references: parabola, center of curvature, equation, slope, plane curve, tangent line, perpendicular, passing through, line, points, curve
There are 28 references to this entry.

This is version 13 of normal line, born on 2007-05-27, modified 2008-03-20.
Object id is 9476, canonical name is NormalLine.
Accessed 4189 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

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