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One of the basic concepts in geometry. The definitions of a surface in various fields of geometry differ substantially.
 
One of the basic concepts in geometry. The definitions of a surface in various fields of geometry differ substantially.
  
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In analytic and algebraic geometry, a surface is considered as a set of points the coordinates of which satisfy equations of a particular form (see, for example, [[Surface of the second order|Surface of the second order]]; [[Algebraic surface|Algebraic surface]]).
 
In analytic and algebraic geometry, a surface is considered as a set of points the coordinates of which satisfy equations of a particular form (see, for example, [[Surface of the second order|Surface of the second order]]; [[Algebraic surface|Algebraic surface]]).
  
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In three-dimensional Euclidean space $  E  ^ {3} $,  
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In three-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913301.png" />, a surface is defined by means of the concept of a surface patch — a homeomorphic image of a square in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913302.png" />. A surface is understood to be a connected set which is the union of surface patches (for example, a sphere is the union of two hemispheres, which are surface patches).
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a surface is defined by means of the concept of a surface patch — a homeomorphic image of a square in $  E  ^ {3} $.  
 
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A surface is understood to be a connected set which is the union of surface patches (for example, a sphere is the union of two hemispheres, which are surface patches).
 
  
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Usually, a surface is specified in $  E  ^ {3} $
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Usually, a surface is specified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913303.png" /> by a vector function
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by a vector function
 
  
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$$
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<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/s/s091/s091330/s0913304.png" /></td> </tr></table>
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\mathbf r  = \mathbf r ( x( u , v), y( u , v), z( u , v)),
 
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$$
 
  
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where $  0 \leq  u , v \leq  1 $,
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913305.png" />, while
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while
 
  
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$$
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<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/s/s091/s091330/s0913306.png" /></td> </tr></table>
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x  = x( u, v),\ \
 
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y  = y( u, v),\ \
 
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z  = z( u, v)
 
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$$
 
  
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are functions of parameters $  u $
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are functions of parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913308.png" /> that satisfy certain regularity conditions, for example, the condition
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and $  v $
 
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that satisfy certain regularity conditions, for example, the condition
 
  
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$$
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<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/s/s091/s091330/s0913309.png" /></td> </tr></table>
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\mathop{\rm rank}  \left \|
 
  
 
(see also [[Differential geometry|Differential geometry]]; [[Theory of surfaces|Theory of surfaces]]; [[Riemannian geometry|Riemannian geometry]]).
 
(see also [[Differential geometry|Differential geometry]]; [[Theory of surfaces|Theory of surfaces]]; [[Riemannian geometry|Riemannian geometry]]).
  
 
From the point of view of topology, a surface is a [[Two-dimensional manifold|two-dimensional manifold]].
 
From the point of view of topology, a surface is a [[Two-dimensional manifold|two-dimensional manifold]].
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====Comments====
 
====Comments====
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979) {{MR|0528129}} {{ZBL|0404.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979) {{MR|0528129}} {{ZBL|0404.53001}} </TD></TR></table>

Revision as of 14:53, 7 June 2020

One of the basic concepts in geometry. The definitions of a surface in various fields of geometry differ substantially.

In elementary geometry, one considers planes, multi-faced surfaces, as well as certain curved surfaces (for example, spheres). Each curved surface is defined in a special way, very often as a set of points or lines. The general concept of surface is only explained, not defined, in elementary geometry: One says that a surface is the boundary of a body, or the trace of a moving line, etc.

In analytic and algebraic geometry, a surface is considered as a set of points the coordinates of which satisfy equations of a particular form (see, for example, Surface of the second order; Algebraic surface).

In three-dimensional Euclidean space , a surface is defined by means of the concept of a surface patch — a homeomorphic image of a square in . A surface is understood to be a connected set which is the union of surface patches (for example, a sphere is the union of two hemispheres, which are surface patches).

Usually, a surface is specified in by a vector function

where , while

are functions of parameters and that satisfy certain regularity conditions, for example, the condition

(see also Differential geometry; Theory of surfaces; Riemannian geometry).

From the point of view of topology, a surface is a two-dimensional manifold.


Comments

References

[a1] J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)
[a2] J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979) MR0528129 Zbl 0404.53001
How to Cite This Entry:
Surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface&oldid=49459
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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