# 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 ?
The film is placed between the occlusal surfaces of the teeth and the patient bites down gently to hold the film in place.
oregonensis also had the thickest cuticle on the occlusal surface of the pollex (1.
The purpose of models composition procedure is to replace occlusal surface of tooth model, created by computed tomography scanning data with a more detailed occlusal surface model, based on optical scanning of gypsum model data.
The teeth are arranged in leaf-like layers; as those on the occlusal surface become worn and are lost, they are replaced by teeth from underlying layers.
Dens evaginatus is one of the developmental anomalies affecting the shape of teeth having a tubercle like extra cusp either on lingual or occlusal surface of teeth.
Posterior approximal caries can be diagnosed with the light probe positioned on the gingivae below the cervical margin of the tooth, whereby the light passes through the tooth structures and approximal decay produces a dark shadow on the occlusal surface.
Ideally, the assistant would position the HVE inlet on the buccal aspect of the teeth that the dentist is working on, with the inlet approximately at a right angle to the occlusal surface of the teeth, or with the inlet slightly occlusal to the occhsal surface and facing the occlusal surface.
Acrylic resin was added on the occlusal surface of this plate with the models mounted in the articulator.
In case the occlusal surface presents unwanted cavities, or pits and fissures, bonding agent (resin) is applied (without acid upon the enamel) in order to create a more favorable occlusal anatomy (fig.
No sigmoid curvature of the occlusal surface of the toothplates is present, as seen from both occlusal and lateral views, thus differentiating them from members of the Phyllodontinae.
The Von Spee's Curve (VSC) is defined as the line on a cylinder tangent to the anterior border of the condyle, the occlusal surface of lower second molar, and the incisal edges of the lower incisors.
Stains and gloss applied on the occlusal surface and fired.

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