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1. Geometry
a. the complete boundary of a solid figure
b. a continuous two-dimensional configuration
a. the uppermost level of the land or sea
b. (as modifier): surface transportation
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.



a fundamental geometric concept with different meanings in different branches of geometry.

(1) A high-school geometry course considers planes, polyhedrons, and some curved surfaces. Each of the curved surfaces is defined in a special way— most often as a set of points that satisfy certain conditions. For example, the surface of a sphere is the set of points at a specified distance from a given point. The concept of a surface is merely exemplified rather than defined. Thus, a surface is said to be the boundary of a solid or the trace of a moving curve.

(2) The mathematically rigorous definition of a surface is based on the concepts of topology. The principal concept here is that of a simple surface, which may be represented as a part of a plane that is subject to continuous deformation— that is, to continuous extension, compression, or bending. More precisely, a simple surface is the image of the interior of a square under a homeomorphic, that is, a one-to-one and bicontinuous, mapping. This definition can be expressed analytically as follows. Introduce Cartesian coordinates u, v in the plane and x, y, z in space. Let S be the (open) square whose points have coordinates satisfying the inequalities 0 < u < 1 and 0 < v < 1. A simple surface is the homeomorphic image in space of the square Sʹ. The surface is given by means of formulas x = Φ (u, v), y = ψ(u, v), z = x(u, v), which are called its parametric equations. For different points (u, v) and (u ʹ, vʹ) the corresponding points (x, y, z) and (xʹ, yʹ, zʹ) must be different, and the functions Φ(u, v), ψ(u, v), and x(u, v) must be continuous. The hemisphere is an example of a simple surface. The sphere, however, is not a simple surface. Further generalization of the concept of a surface is consequently necessary. If a neighborhood of each point of a surface is a simple surface, the surface is said to be regular. From the standpoint of topological structure, surfaces as twodimensional manifolds are divided into several types, such as closed and open surfaces and orientable and nonorientable surfaces.

The surfaces investigated in differential geometry usually obey conditions associated with the possibility of using the methods of the differential calculus. These are usually smoothness conditions, such as the existence of a tangent plane or of curvature at each point of the surface. These requirements mean that the functions Φ(u, v), ψ(u, v), and x (u, v) are assumed to be once, twice, three times, or, in some problems, infinitely differentiable or even analytic. Moreover, it is required that at each point at least one of the determinants

be nonzero.

In analytic and algebraic geometry, a surface is defined as a set of points whose coordinates satisfy an equation of the form

(*) Φ(x, y, z) = 0

Thus, a given surface may or may not have a graphic geometric image. In this case, in order to preserve generality, we speak of imaginary surfaces. For example, the equation

X2 + y2 + z2 + 1 = 0

defines an imaginary sphere, although real space contains no point with coordinates satisfying this equation. If the function Φ(x, y, z) is continuous at some point and has at this point continuous partial derivatives ∂Φ/ ∂x, ∂Φ/ ∂y, ∂Φ/∂z, at least one of which does not vanish, then in the neighborhood of this point the surface defined by equation (*) will be a regular surface.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.


The outer part (skin with a thickness of zero) of a body; can apply to structures, to micrometer-sized particles, or to extended-surface zeolites.
A subset of three-space consisting of those points whose cartesian coordinates x, y, and z satisfy equations of the form x = ƒ(u, v), y = g (u, v), z = h (u, v), where ƒ, g, and h are differentiable real-valued functions of two parameters u and v which take real values and vary freely in some domain.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.


(1) (Surface) Microsoft's hardware brand. See Surface versions.

(2) In CAD, the external geometry of an object. Surfaces are generally required for NC (numerical control) modeling rather than wireframe or solids.
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References in periodicals archive ?
There are dental surfaces where caries develop more frequently [2, 3], including the occlusal surfaces [3,4].
In Case 2, this involved a period of several years--the occlusal surface of the tooth was protected with a fissure sealant.
The occlusal surfaces of some teeth in the specimen are exceptionally flat and possess low, thin transverse ridges that extend fully to the marginal area.
Molars and premolars have large, prominent cusps along the occlusal surface while the canines have only small cusps on their incisal edges.
The most frequently affected areas are the palatal surface of the maxillary incisors and the occlusal surface of the mandibular first molars in adolescents [15-17].
A maxillary cross-sectional occlusal radiograph was showing a solitary diffuse radiolucency approximately of size 4 x 2 cm in the right maxilla extending anterio-posteriorly from interdental areas of 11, 12, and 53 to horizontal line drawn from the occlusal surface of 17.
The only non significant (p=0.112) difference in the dimensions was that of non functional cusps of metal ceramic crowns which may be due to multiple factors one of which is over prepared flat occlusal surface. Flat occlusal surface causes adequate or over reduction of non function cusp whereas under reduction of functional cusp.
Another topological association studied was that between the level of the occlusal surface of the second molar and the level of point F referred to as height of point F (Hf).
Acrylic resin was added on the occlusal surface of this plate with the models mounted in the articulator.