In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface.
Other articles related to "surface, surfaces":
... In a hard drive, the heads 'fly' above the disk surface with clearance of as little as 3 nanometres ... head is controlled by the design of an air-bearing etched onto the disk-facing surface of the slider ... height constant as the head moves over the surface of the disk ...
... Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry ... It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E² ... This elaboration allows calculus to be applied to surfaces to prove many results ...
... surveying and geodesy, a datum is a set of reference points on the Earth's surface against which position measurements are made and (often) an associated model of the shape of the Earth (reference ellipsoid ... Horizontal datums are used for describing a point on the Earth's surface, in latitude and longitude or another coordinate system ... In engineering and drafting, a datum is a reference point, surface, or axis on an object against which measurements are made ...
... The field's surface, originally composed of AstroTurf, contained many gaps and uneven patches ... Baseball players also complained about the surface ... It was much harder than other AstroTurf surfaces, and the shock of running on it often caused back pain ...
... The Roman surface or Steiner surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane ... and latitude (φ), gives parametric equations for the Roman surface as follows x = r2 cos θ cos φ sin φ y = r2 sin θ cos φ sin φ z = r2 cos θ sin θ cos2 φ ... is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there ...
Famous quotes containing the word surface:
“A novelist is, like all mortals, more fully at home on the surface of the present than in the ooze of the past.”
—Vladimir Nabokov (18991977)
“See how peaceful it is here. The sea is everything. An immense reservoir of nature where I roam at will.... Think of it. On the surface there is hunger and fear. Men still exercise unjust laws. They fight, tear one another to pieces. A mere few feet beneath the waves their reign ceases, their evil drowns. Here on the ocean floor is the only independence. Here I am free.”
—Earl Felton, and Richard Fleischer. Captain Nemo (James Mason)
“These wonderful things
Were planted on the surface of a round mind that was to become our present time.”
—John Ashbery (b. 1927)