# 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) See Surface tablet, Surface Book 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 ?
where [PHI] are values from the DI map in 62 body surface points, [Ti.
Body surface potential mapping: This noninvasive method comprises 87 electrocardiographic electrodes to be distributed 58 on the anterior, and 29 on the posterior surface of the body.
Key words: single-vessel disease, resting ECG, body surface potential mapping, ST-T isointegral maps
The most general advantage of body surface potential maps (BSPM) is the fact that they record cardiac potentials from broad areas of the chest, thus enabling the detection of significant physiologic and possibly diagnostic information outside the precordial regions usually explored.
Additionally women generally have less body surface area than men.
As in other mammals, there are multiple representations of the body surface in this brain region, report M.
Siemens Tim technology is the first seamless, whole body surface coil design that combines up to 102 seamlessly integrated coil elements with up to 32 independent radio frequency channels, opening the door to the most advanced clinical applications available today.
There is also a longitudinal medial hump on the body surface in the longitudinal central region of the sanitary napkin, which has a point of maximum amplitude and a caliper measured at its point of maximum amplitude that is greater than about 3 mm, which is at least about two times the caliper of the portions of the longitudinal side regions.
Siemens Tim technology is the first seamless, whole body surface coil design that combines up to 102 seamlessly integrated coil elements with up to 32 independent RF Channels, opening the door to the most advanced clinical applications available today.
The main front bend line and the two rear-wardly diverging bend lines fold so that the portions of the body surface on opposi te sides of the main front bend line and the two rear-wardly diverging bend lines are brought closer together to provide a recessed portion along the main front bend line adapted to be remote from a user's body.
Built-in medical calculators that provide instant access from within topics to relevant medical calculators including: Body Surface Area: Pediatric; Creatinine Clearance - Measured; Fick - Cardiac Output; Glasgow Coma Scale; Head Circumference: 0-36 Months; Ideal Body Weight (IBW); Stature:Weight Ratio; Water Deficit; and more.
The absorbent article also contains a liquid pervious body surface, a liquid impervious garment surface, an absorbent core positioned between the body and garment surfaces and a fastener on the garment surface for fastening the absorbent article to the crotch region of a wearer s undergarment.

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