# 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
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## 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.
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References in periodicals archive ?
Positioning of the crowns with the flattened facial surface parallel to the vertical stem of the surveyor with pink wax number 7 (Figure 2).
From top to bottom: bare facial surface, shape index S, second fundamental form f and derivative along x.
In addition, an impression of the facial surfaces of the unprepared anterior teeth can be made to show how the anterior teeth look, by having the patient occlude the teeth, and placing fast-setting bite registration material on the facial surfaces of both the mandibular and maxillary anterior teeth (Fig.
The core of our 3D face recognition framework is the representation of a facial surface which is invariant to isometric deformations, by bending invariants (BI) [16, 17].
Using only the chosen colours--quite a discipline in itself--Cazalet often employed quite strong geometrical shapes to highlight the facial surfaces and features, making a reference to theatrical make-up and the emphasis this can give to both facial and inner characteristics.
(18-20) The Modified Bass method requires that the toothbrush bristles be placed at a 45-degree angle to the gingival margin and moved gently back and forth followed by a roll or sweeping motion across the broad lingual or facial surfaces to clean the remainder of the tooth.
The color was recorded using the Vita Classical shade guide, and the shade A2 and A3 was considered as the initial color the tooth preparation involved a minimal chamfer in the facial surfaces. The enamel surface was acid etched using 35% phosphoric acid gel (Ultra-Etch; Ultradent, South Jordan, UT, USA) for 15 s, rinsed for 10 s and dried.
According to the study, by the way, you want to brush&nbsp;in a circular motion, use a 45-degree angle while brushing facial surfaces of teeth and position your arm correctly (theses were some of the criteria for the "tooth-brushing skill" metric).
These 3D data sets showed individual brain regions, as well as accurate and detailed facial surfaces, from which Hammond and research assistant and co-author Michael Suttie performed shape analyses.
His chapter titles usefully indicate the precision of his analysis: "Losing Face in Typee," "Jumping out of One's Skin in White-Jacket," "Getting inside Heads in Moby-Dick," "Penetrating Eyes in Pierre," and "Inscribed Hearts in Pierre." We can best understand Tommo's anxiety about having his face tattooed, Otter suggests, if we set it in the context of the invidiously comparative mappings of facial surfaces in the linked disciplines of phrenology and racial ethnology.
The extended pieces allowed the masks to be displayed without suffering any damage to facial surfaces.
The tooth preparation involved a minimal chamfer on the facial surfaces. Cotton rolls and salivary ejectors were used for field isolation.

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