# Normal

*to a curve (or surface) at a point of it*

A straight line passing through the point and perpendicular to the tangent (or tangent plane) of the curve (or surface) at this point. A smooth plane curve has at every point a unique normal situated in the plane of the curve. If a curve in a plane is given in rectangular coordinates by an equation , then the equation of the normal to the curve at has the form

A curve in space has infinitely many normals at every point of it. These fill a certain plane (the normal plane). The normal lying in the osculating plane is called the principal normal; the one perpendicular to the osculating plane is called the binormal.

The normal at to a surface given by an equation is defined by

If the equation of the surface has the form , then the parametric representation of the normal is

#### Comments

The notion of a normal obviously extends to -dimensional submanifolds of Euclidean -space , giving an -dimensional affine subspace as the normal -plane to the manifold at the corresponding point. For submanifolds of (pseudo-) Riemannian manifolds, the normal planes are considered as subspaces of the tangent space of the ambient space, where orthogonality is defined by means of the (ambient) (pseudo-) Riemannian metric. See also Normal bundle; Normal plane; Normal space (to a surface).

#### References

[a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |

[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |

[a3] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145 |

[a4] | M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5 |

[a5] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |

[a6] | B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973) |

**How to Cite This Entry:**

Normal.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Normal&oldid=11310