# 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 |