Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space. For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections (it cannot be embedded in the three dimensional Euclidean space).

In mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R³, such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.

A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (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 is widely used 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.

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