surface

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Related to occlusal surface: occlusal wear, occlusal relationship, occlusal load, proximal 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) See Surface tablet, Surface Book, Surface Studio and PixelSense.

(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 ?
Various methods may be used to minimize model's descriptions, though it is necessary to keep suitably detailed occlusal surface, particularly in contact zones of occlusal surfaces.
The assistant would move to the patient's right side and position the HVE shaft buccal to the teeth being worked on, with the inlet facing the occlusal surfaces of the teeth.
Another hypothesis for the results is that a single application of the fluoridated agent on the occlusal surface does not enhance enamel acid resistance.
Of the seven toothplates, six have a convex occlusal surface whereas the seventh is concave.
2010] studied the validity of three different methods for fissure caries detection in 144 second molars with intact occlusal surfaces in 41 young adults, including visual examination, bitewing radiographs, and the use of a DIAGNOdent.
The plaque and debris on the occlusal surfaces were removed using a rotating brush without prophylaxis paste.
The first was cut in the middle of the occlusal surface, and the other two were cut 2 mm to the left and 2 mm to the right of the central section, respectively (Figure 1)
To produce suitable specimens, all teeth were mounted in plaster leaving only their occlusal surface free.
Clinical examination revealed a normal mixed dentition in all quadrants with the exception of the mandibular right side where the first permanent molar was severely submerged, its occlusal surface being at the gingival level.
East Windsor, NJ, USA) was used as the sealant material, applied on fully erupted teeth (entire occlusal surface was exposed into the mouth with no overlying tissue operculum).