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 It might be outdated or ideologically biased.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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

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.
Copyright © 1981-2019 by The Computer Language Company Inc. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.
References in periodicals archive ?
in which [r.sub.cnd/csf] = [r.sub.cnd] - [r.sub.csf] is the contact node's relative position with respect to the contact surface frame.
Caption: FIGURE 18: The diagrams of the contact surface and the calculation points.
Calculation of movement of the contact surface reduces to a multiplication of the interference maximum sequence number the numerical value of the wavelength of the radiation source (in this case 0.3162 [micro]m).
Figure 6 shows the normal contact stress distribution on the contact surface, when the interferences are, respectively, 0.003 mm, 0.006 mm, and 0.01 mm.
This cadaveric study was able to quantify the contact surface area of the LHB with the humeral articular surface and qualitatively describe the pathway of the LHB.
Components in many rotary devices and in automated processes transfer heat periodically across contact surfaces. Example includes heat transfer between a soldering iron and work piece on an assembly line.
In the figure 5 is shown the roughness profile on the contact surface of the initial workpiece.
The upper case is lowered onto the bedplate, and a liquid sealing compound is injected into a groove in the contact surface.
Also introduced is the new PS chipbreaker with a wide contact surface for stability in the pocket and a land that provides good balance of sharpness and durability.
After polymerisation, a thin slice was cut with a diamond saw from the soil root contact between 0.5 and 1.0 cm thick, transverse to the root soil contact surface. The samples were mounted on glass slides (2 by 6cm) using araldite glue and ground to a thickness of 30 [micro]m.
The most common method of controlling [C.sub.RES] uses abrasive cleaning to physically remove the contaminants from the contact surface. For this type of cleaning, the probes are subjected to multiple touchdowns on lapping film with 0.5-, 1-, 3-, or 5-[micro]m grit; a tungsten-carbide plate; or a ceramic block.