View source for Surface

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|Surface of the second order]]; [[Algebraic surface|Algebraic surface]]).

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).

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

<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091330/s0913305.png" />, 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 <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

<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]]).

From the point of view of topology, a surface is a [[Two-dimensional manifold|two-dimensional manifold]].



====Comments====


====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>