# 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; 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.

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#### References

[a1] | J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969) |

[a2] | J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979) |

**How to Cite This Entry:**

Surface.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Surface&oldid=11661