# 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 It might be outdated or ideologically biased.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 ?
Osteochondral fracture is a common injury following patellar dislocation, of which the incidence is reported from 31% to 58% of acute patellar dislocation.[4],[5],[6],[7] An ideal decision of treating the osteochondral fracture should be made according to injury pattern including the location and the size of fracture and whether bearing surface is involved or not.
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Following unsuccessful attempts that included interposition of materials such as ivory, gold foil, glass, stainless-steel, acrylic and ceramic between the joint bearing surfaces to relieve pain, in 1938 Philip Wiles at the Middlesex Hospital, London undertook the first prosthetic joint replacement (Wiles 1958).
It was reported that impingement between the femoral neck and acetabular cup could presumably be caused by malposition, or that improper design of the implants could produce third bodies.[sup][11],[39] Metal debris from impingement could fall inside the bearing surface and disrupt the lubrication film, which would bring about an increased wear rate of the ceramic surfaces, thus producing ceramic debris.
The few times I have had to reduce bearing surface was when dealing with freeze-off at the tip orifice, requiring excessive pressure to push plastic through--if it was possible at all.
* A gradual transition section (a variable pitch between the last coil at full pitch and the dead coil, on both ends) maximizes square/flat load bearing surface while maintaining a smooth/kink-free design.
(6-8) Yet even with reports of no aseptic loosening at 10 or 15 years post-implantation, wear of the bearing surface remains a serious threat to not only the long-term stability of the implant but to bone loss should revision arthroplasty be needed.
"Surgeons indicated that clinical results and patient needs had the highest impact on their choice of bearing surface," noted Andrea Cheng, a senior analyst at MRG.
Bearing makers index bearing surface speeds by multiplying bearing diameter times rotation rate for a number they call DN.
One issue not mentioned in Mike's article is bullet bearing surface length and case neck length.

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