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

In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness. Image File history File links Download high resolution version (1000x1000, 499 KB) Shows a saddle point on a surface between two hills. ... Image File history File links Download high resolution version (1000x1000, 499 KB) Shows a saddle point on a surface between two hills. ... Euclid, detail from The School of Athens by Raphael. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... Rain falling Rain on an umbrella Rain is a form of precipitation, as are snow, sleet, hail, and dew. ... A soap bubble A soap bubble is a very thin film of soap water that forms a hollow shape with an iridescent surface. ... Snow crystal A snowflake is an aggregate of snow crystals that form while falling in and below a cloud. ... In physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. ... Surface chemistry is the study of chemical phenomena that occur at the interface of two phases, usually between a gas and a solid or between a liquid and a solid. ... Surface energy quantifies the disruption of chemical bonds that occurs when a surface is created. ... Roughness or rugosity is a measurement (see surface metrology) of the small-scale variations in the height of a physical surface. ...


The two-dimensional character of a surface comes from the fact that, about each point, there is a "coordinate patch" on which a two-dimensional coordinate system is defined; in general, it is not possible to extend this patch to the entire surface, so it will be necessary to define multiple patches which collectively cover the surface. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...


A surface may have a boundary, where the surface ends. For example, the boundary of a disc or hemisphere would be the circle around the edge. In geometry, a disk is the region in a plane contained inside of a circle. ... A sphere is a perfectly symmetrical geometrical object. ...

Contents


Examples

The general concept of a surface, and the richness and variety of surfaces, can be understood by examining a variety of examples. Any formal definition of a surface must be strong enough to encompass this variety.

Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ... A developable surface is a surface that can be flattened onto a plane without distortion (i. ... Look up cylinder in Wiktionary, the free dictionary. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... A torus. ... In geometry, a surface is ruled if through every point of there is a straight line that lies on . ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ... A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of revolution) that lies on the same plane. ... Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ... A soap film is a physical realization of a minimal surface. ... A catenoid A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. ... The helicoid is one of the first minimal surfaces discovered. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... The word locus (plural loci) is Latin for place: In biology and evolutionary computation, a locus is the position of a gene (or other significant sequence) on a chromosome. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... A cubic surface is a projective variety studied in algebraic geometry. ... In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space. ... In mathematics and computer graphics, an implicit surface is defined as an isosurface of a function . In other words, it is the set of points in the 3d-space that satisfy the equation . To find a parametrisation of the surface (more precisely the solution set, since not all equations of... The Klein bottle immersed in three-dimensional space. ... A Möbius strip made with a piece of paper and tape. ... // Orientability of surfaces The torus is an orientable surface. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... A sphere is a perfectly symmetrical geometrical object. ... A torus. ... In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from the combinatorics of block designs. ... In mathematics, a projective space is a fundamental construction from any vector space. ... In geometry, Boys surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. ... The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. ... In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. ... The Steiner surfaces are self-intersecting embeddings of the real projective plane into three-dimensional space. ... A drawing of Alexanders horned sphere Alexanders Horned Sphere is one of the most famous pathological examples in mathematics. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

Definition

In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second countable if its topology has a countable base. ...


More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E2 (Euclidean 2-space) or an open subset of the closed half of E2. The set of points which have an open neighbourhood homeomorphic to En is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1)-manifold, or union of closed curves. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


A surface with empty boundary is said to be closed if it is compact, and open if it is not compact. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...


Classification of closed surfaces

There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of two infinite collections: In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... The word Boundary has a variety of meanings. ...

Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism). A sphere is a perfectly symmetrical geometrical object. ... This article or section should be merged with Orientable manifold. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... It has been suggested that this article or section be merged with orientable manifold. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...


Compact surfaces

Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint. In geometry, a disk is the region in a plane contained inside of a circle. ...


Embeddings in R3

A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4. In differential topology, the Whitney embedding theorem states that Any smooth second-countable -dimensional manifold can be embedded in Euclidean -space. ...


Differential geometry

A simple review of the embedding of a surface in n dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the article Poincaré metric. In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. ...


Some models

To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows match:

Fundamental polygon

Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...


This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon.


The above models can be described as follows:

  • sphere: AA − 1
  • projective plane: AA
  • Klein bottle: ABA − 1B
  • torus: ABA − 1B − 1

(See the main article fundamental polygon for details.) In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...


Connected sum of surfaces

Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components.


We use the following notation.

  • sphere: S
  • torus: T
  • Klein bottle: K
  • Projective plane: P

Facts:

  • S # S = S
  • S # M = M
  • P # P = K
  • P # K = P # T

We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S.


Closed surfaces are classified as follows:

  • gT (g-fold torus): orientable surface of genus g, for g ge 0.
  • gP (g-fold projective plane): non-orientable surface of genus g, for g ge 1.

Algebraic surface

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a real manifold. Algebraic surfaces over the real numbers will give normal surfaces, however these may contain singular points, where the algebraic surface forms a degenerate lines or points. In mathematics, an algebraic surface is an algebraic variety of dimension two. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...


External links

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