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A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an open disk (which is a sphere with a puncture), an open cylinder (which is a sphere with two punctures), and the Möbius strip. A surface embedded in three-dimensional space is closed if and only if it is the boundary of a ...
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.
A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces. These include analogs to physical ...
Simple examples. A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
This is a list of surfaces in mathematics. They are divided into minimal surfaces , ruled surfaces , non-orientable surfaces , quadrics , pseudospherical surfaces , algebraic surfaces , and other types of surfaces.
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface =
Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces; Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane