# surface

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## surface

1. Geometry
a. the complete boundary of a solid figure
b. a continuous two-dimensional configuration
2.
a. the uppermost level of the land or sea
b. (as modifier): surface transportation

## Surface

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.

## surface

[′sər·fəs]
(engineering)
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.
(mathematics)
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.

## surface

(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.
References in periodicals archive ?
Microtubules gradually become more focused on the four tetrahedrally arranged plastids, first around their entire surfaces and then concentrated on the proximal surfaces (Fig.
Pearson Correlation Coefficient (r) was used to determine the grade of correlation between measurements of buccal/palatal and proximal surfaces. Cutoff points were > 0.70: strong correlation; 0.30 to 0.70: moderate correlation; < 0.30: weak correlation.
Tight contact points between the proximal surfaces of the primary molars could be associated with an increased risk of the presence and activity of proximal caries, (1-3) as the initiation and progression of proximal caries lesions are related to higher plaque accumulation in these conditions.
Gingival bleeding was assessed by flossing all proximal surfaces [Caton and Polson, 1975; Tinoco and Gjermo, 1992].
An evaluation of proximal surface cleansing agents.
Moreover, there is enough clinical evidence that states the effectiveness of this procedure both on occlusal (7,8,10,11,17) and proximal surfaces. (3,4) A recent review on the effect of dental sealants on the bacteria levels in caries lesions found that sealants significantly reduced bacteria levels in cavitated lesions, (5) supporting the findings of a recent meta-analysis that sealants prevented caries progression.
[11.] Kantor ML, Reiskin AB, Lurie AG: A clinical comparison of x-ray films for the detection of proximal surface caries.
The radiographs were scanned to obtain digitalized images to measure the crestal bone level (CBL that is the distance from cementoenamel junction (CEJ) to the alveolar crest) on proximal surfaces using computerized software (Figure 3).
These types of studies have been completed primarily in children and show a reduction of caries predominantly on proximal surfaces. There are several different polyol sweeteners used in gums and lozenges.
Ten-year-olds with foreign backgrounds had statistically significantly more DFS on proximal surfaces in 2003.
Data on the circumference of the proximal metaphysis were collected from 44 tibiae exhibiting various degrees of degeneration of the proximal surfaces. Statistical analysis indicates that the broadness of the proximal metaphysis is positively correlated with the degree of observable joint degeneration as scored by Jurmain's (1975) method.

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