# 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 ?
Sometimes, the dentist may move to the patient's left to get a direct vision view of the buccal surfaces of the mandibular posteriors.
Microleakage [n out of 10] Fissure of Buccal surface 0 1 2 3 P Without bevel 4 1 4 1 <0.
Matrix tablets, each containing 18 mg of sodium fluoride, were bonded to the buccal surface of the first maxillary molar teeth of 20 subjects (age 20-23 years).
All the brackets were bonded on the buccal surfaces according to the instructions supplied by the manufacturer.
It was concluded that the appropriate placement site for the electrode of electrical pulp is the Tip of buccal cusp, and the response of the threshold is increased as the electrode is moved apically to the cervical third of the buccal surface of the tooth.
One possible hypothesis is that non-carious class V lesions could occur when erosive dietary fluids (which affect buccal surfaces more than lingual surfaces) leak into the microscopic cracks in the hydroxyapatite that resulted from occlusal loads concentrated into the cervical areas.
Similarly, since lingual surfaces (surface of tooth facing the tongue) are in more contact with saliva than buccal surfaces (surface of tooth facing the buccal and labial mucosa) (24,25), this hypothesis also predicts that the lingual surfaces would be less affected by reduced salivary flow.
As controls, swabs were taken from the buccal surfaces of both volunteers.
2 Study measured thoroughness as the time spent brushing lingual vs buccal surfaces and amount of time spent brushing in each quadrant.
Orthodontic brackets (Orthometric Orthodontics[TM], Marilia, Brazil) were aligned and bonded on the dental buccal surfaces by using a conventional direct bonding technique with the Ortholink Bond System (Orthometric Orthodontics[TM], Marilia, Brazil).
A clinical condition of either Case Type II Early Periodontitis, according to the American Dental Association (ADA) classification system, (24) and supragingival calculus covering the lingual surfaces of the mandibular anterior teeth and the buccal surfaces of the maxillary first molars with subgingival calculus ledges or rings
c) PD of buccal surfaces versus lingual surfaces; and

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