normal line

A normal line (or simply normal or perpendicular) 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 point $P$ is the foot of the normal.

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  $f^{\prime }(x_0)$  and the slope of the normal at that point is  $-{\displaystyle \frac{1}{f^{\prime }(x_0)}}$.  The equation of the normal is thus

$$y-f(x_0)=-\frac{1}{f^{\prime }(x_0)}(x-x_0).$$

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

$$x=x_0,$$

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

$$y=f(x_0).$$

The normal of a curve at its point $P$ always goes 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 $P$.

Title	normal line
Canonical name	NormalLine
Date of creation	2013-03-22 17:09:53
Last modified on	2013-03-22 17:09:53
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	17
Author	pahio (2872)
Entry type	Definition
Classification	msc 26B05
Classification	msc 26A24
Classification	msc 53A04
Synonym	normal of curve
Synonym	normal
Synonym	perpendicular
Related topic	ConditionOfOrthogonality
Related topic	ParallelCurve
Related topic	SurfaceNormal
Related topic	Grafix
Related topic	NormalOfPlane
Defines	foot of normal
Defines	foot of perpendicular