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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 $y=f(x)$, then the equation of the normal to the curve at $(x_0,y_0)$ has the form

$$(x-x_0)+(y-y_0)f'(x_0)=0.$$

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 $(x_0,y_0,z_0)$ to a surface given by an equation $z=f(x,y)$ is defined by

$$\begin{cases}(x-x_0)+(z-z_0)\frac{\partial z}{\partial x}=0,\\(y-y_0)+(z-z_0)\frac{\partial z}{\partial y}=0.\end{cases}$$

If the equation of the surface has the form $\mathbf r=\mathbf r(u,v)$, then the parametric representation of the normal is

$$\mathbf R=\mathbf r+\lambda[\mathbf r_u,\mathbf r_v].$$


Comments

The notion of a normal obviously extends to $m$-dimensional submanifolds of Euclidean $n$-space $E^n$, giving an $(n-m)$-dimensional affine subspace as the normal $(n-m)$-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://www.encyclopediaofmath.org/index.php?title=Normal&oldid=32593
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article