Surface::surfaces    Which::closed    Sphere::space    Boundary::plane    Torus::compact    '''P'''::complex


An open surface with X-, Y-, and Z-contours shown.

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, such as a sphere. 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.

Surface sections
Intro  Definitions and first examples  Extrinsically defined surfaces and embeddings  Construction from polygons  Connected sums   Closed surfaces   Surfaces in geometry  See also  Notes  References  External links  

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