# surface

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Related to lingual surface: buccal surface, distal surface

## 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 ?
Of the three studies that evaluated gingival index (Pandis, (34) Cerny, (35) and Booth (37)), only one (Booth (37)) found statistically significant differences in terms of gingival index scores for maxillary lingual surfaces of both anterior and posterior teeth, with the highest scores in patients with a mandibular retainer in place.
As previously mentioned, it holds only one replacement tooth per tooth position, the labial surface of the crown is thickly enameled and has a strongly denticulated margin with a very large primary ridge arising from the base of the crown, on the lingual surface, the teeth do not display any ornamentation.
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Assessment of the blood supply to the lingual surface of the mandible for reduction of bleeding during implant surgery.
Intrusion of Anterior Teeth: Lingual bracket position, which is dictated by the morphology of the lingual surface of the tooth, places the bracket closer to the center of resistance of the tooth than is found with labial bracket placement and thus clinical implication of this unique bracket position and design is that the intrusive force vector is directed through the center of resistance of the tooth.
* They clinically presented with at least one large class II primary molar carious lesion not extending pre-operatively to the buccal and/or lingual surface.
Moreover, alterations in the CAD/CAM design may consider its bonding to the lingual surface of lower canines only enhancing a sound periodontal status.
The incisal edge is slightly reduced & the finish line terminates on lingual surface.
Most commonly found buccally rotated teeth are premolars with buccal surface facing mesially and lingual surface distally.
Another potential cause for higher failure in the buccal and palatal surfaces could be the enamel rods direction at those surfaces as in buccal or lingual surface cavity preparation; only the sides of the enamel rods are exposed.
All of the findings described above may be the result of environmental adaption and evolution; moreover, the knowledge regarding the root structures of filiform papillae can lay a foundation for future research on the characteristics and functions of lingual surface in ruminants.

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