# normal line

A (or simply normal or ) 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