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Homenormal line

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# 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 $P$.

## Mathematics Subject Classification

26B05*no label found*26A24

*no label found*53A04

*no label found*

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