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''to a curve (or surface) at a point of it''
 
''to a curve (or surface) at a point of it''
  
A straight line passing through the point and perpendicular to the [[Tangent|tangent]] (or [[Tangent plane|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673801.png" />, then the equation of the normal to the curve at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673802.png" /> has the form
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A straight line passing through the point and perpendicular to the [[Tangent|tangent]] (or [[Tangent plane|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673803.png" /></td> </tr></table>
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$$(x-x_0)(y-y_0)f'(x_0)=0.$$
  
 
A curve in space has infinitely many normals at every point of it. These fill a certain plane (the [[Normal plane|normal plane]]). The normal lying in the [[Osculating plane|osculating plane]] is called the [[Principal normal|principal normal]]; the one perpendicular to the osculating plane is called the [[Binormal|binormal]].
 
A curve in space has infinitely many normals at every point of it. These fill a certain plane (the [[Normal plane|normal plane]]). The normal lying in the [[Osculating plane|osculating plane]] is called the [[Principal normal|principal normal]]; the one perpendicular to the osculating plane is called the [[Binormal|binormal]].
  
The normal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673804.png" /> to a surface given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673805.png" /> is defined by
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The normal at $(x_0,y_0,z_0)$ to a surface given by an equation $z=f(x,y)$ is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673806.png" /></td> </tr></table>
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$$\begin{cases}(x-x_0)+(z-z_0)\frac{\partial z}{\partial x}=0,\\(y-y_0)+(z-z_0)\frac{\partial z}{\partial y}=0.\end{cases}$$
  
If the equation of the surface has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673807.png" />, then the parametric representation of the normal is
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If the equation of the surface has the form $\mathbf r=\mathbf r(u,v)$, then the parametric representation of the normal is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673808.png" /></td> </tr></table>
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$$\mathbf R=\mathbf r+\lambda[\mathbf r_u,\mathbf r_v].$$
  
  
  
 
====Comments====
 
====Comments====
The notion of a normal obviously extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n0673809.png" />-dimensional submanifolds of Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n06738010.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n06738011.png" />, giving an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n06738012.png" />-dimensional affine subspace as the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067380/n06738013.png" />-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 bundle]]; [[Normal plane|Normal plane]]; [[Normal space (to a surface)|Normal space (to a surface)]].
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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 bundle]]; [[Normal plane|Normal plane]]; [[Normal space (to a surface)|Normal space (to a surface)]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR></table>

Revision as of 17:14, 30 July 2014

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://encyclopediaofmath.org/index.php?title=Normal&oldid=32592
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article