Difference between revisions of "Surface"
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
Ulf Rehmann (talk | contribs) m (Undo revision 48914 by Ulf Rehmann (talk)) Tag: Undo |
||
Line 1: | Line 1: | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
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. | ||
Line 17: | Line 5: | ||
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]]). | ||
− | In three-dimensional Euclidean space | + | 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). |
− | 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 | + | 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 |
− | by a vector function | ||
− | + | <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> | |
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913305.png" />, while |
− | while | ||
− | + | <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> | |
− | |||
− | |||
− | |||
− | |||
− | are functions of parameters | + | 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 |
− | and | ||
− | that satisfy certain regularity conditions, for example, the condition | ||
− | + | <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> | |
− | |||
(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]]. | ||
+ | |||
+ | |||
====Comments==== | ====Comments==== | ||
+ | |||
====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 |
Surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface&oldid=48914