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{{short description|Mathematical idealization of the surface of a body}}
[[Image:Sphere and Ball.png|right|thumb|A [[Sphere|sphere]] is the surface of a solid [[Ball (mathematics)|ball]], here having [[Radius|radius]] ''r'']]

In [[Mathematics|mathematics]], a '''surface''' is a [[Mathematical model|mathematical model]] of the common concept of a [[Surface|surface]]. It is a generalization of a [[Plane (mathematics)|plane]], but, unlike a plane, it may be [[Curvature|curved]]; this is analogous to a [[Curve|curve]] generalizing a straight line. 

There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and [[Sphere|sphere]]s in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in [[Algebraic geometry|algebraic geometry]], a surface may cross itself (and may have other [[Singular point of an algebraic variety|singularities]]), while, in [[Topology|topology]] and [[Differential geometry|differential geometry]], it may not.

A surface is a [[Topological space|topological space]] of [[Physics:Dimension|dimension]] two; this means that a moving point on a surface may move in two directions (it has two [[Degrees of freedom|degrees of freedom]]). In other words, around almost every point, there is a ''coordinate patch'' on which a [[Two-dimensional space|two-dimensional]] [[Coordinate system|coordinate system]] is defined. For example, the surface of the Earth resembles (ideally) a sphere, and [[Earth:Latitude|latitude]] and [[Earth:Longitude|longitude]] provide two-dimensional coordinates on it (except at the poles and along the [[Earth:180th meridian|180th meridian]]).

==Definitions==
Often, a surface is defined by [[Equation|equation]]s that are satisfied by the coordinates of its points. This is the case of the [[Graph of a function|graph]] of a [[Continuous function|continuous function]] of two variables. The set of the [[Zero of a function|zeros of a function]] of three variables is a surface, which is called an [[Implicit surface|implicit surface]].<ref>Here "implicit" does not refer to a property of the surface, which may be defined by other means, but instead to how it is defined. Thus this term is an abbreviation of "surface defined by an implicit equation".</ref> If the defining three-variate function is a [[Polynomial|polynomial]], the surface is an [[Algebraic surface|algebraic surface]]. For example, the [[Unit sphere|unit sphere]] is an algebraic surface, as it may be defined by the implicit equation
:<math>x^2+y^2+z^2 -1= 0.</math>

A surface may also be defined as the [[Image (mathematics)|image]], in some space of [[Physics:Dimension|dimension]] at least 3, of a [[Continuous function|continuous function]] of two variables (some further conditions are required to insure that the image is not a [[Curve|curve]]). In this case, one says that one has a [[Parametric surface|parametric surface]], which is ''parametrized'' by these two variables, called ''parameters''. For example, the unit sphere may be parametrized by the [[Euler angles]], also called [[Earth:Longitude|longitude]] {{mvar|u}} and [[Earth:Latitude|latitude]] {{mvar|v}} by 
:<math>\begin{align}
x&= \cos(u)\cos(v)\\
y&=\sin(u)\cos(v)\\
z&=\sin(v)\,.
\end{align}</math>

Parametric equations of surfaces are often irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles ([[Modulo operation|modulo]] {{math|2{{pi}}}}). For the remaining two points (the [[Earth:North Pole|north]] and [[Earth:South Pole|south poles]]), one has {{math|1=cos ''v'' = 0}}, and the longitude {{math|''u''}} may take any values. Also, there are surfaces for which there cannot exist a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations, whose images cover the surface. This is formalized by the concept of [[Manifold|manifold]]: in the context of manifolds, typically in [[Topology|topology]] and [[Differential geometry|differential geometry]], a surface is a manifold of dimension two; this means that a surface is a [[Topological space|topological space]] such that every point has a neighborhood which is [[Homeomorphism|homeomorphic]] to an open subset of the [[Euclidean plane]] (see [[Surface (topology)]] and Surface (differential geometry)). This allows defining surfaces in spaces of dimension higher than three, and even ''abstract surfaces'', which are not contained in any other space. On the other hand, this excludes surfaces that have [[Singularity theory|singularities]], such as the vertex of a [[Conical surface|conical surface]] or points where a surface crosses itself.

In classical geometry, a surface is generally defined as a [[Locus (mathematics)|locus]] of a point or a line. For example, a [[Sphere|sphere]] is the locus of a point which is at a given distance of a fixed point, called the center; a [[Conical surface|conical surface]] is the locus of a line passing through a fixed point and crossing a [[Curve|curve]]; a [[Surface of revolution|surface of revolution]] is the locus of a curve rotating around a line. A [[Ruled surface|ruled surface]] is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is a [[Union (set theory)|union]] of lines.

==Terminology==
There are several kinds of surfaces that are considered in mathematics. An unambiguous terminology is thus necessary to distinguish them when needed. A ''topological surface'' is a surface that is a [[Manifold|manifold]] of dimension two (see {{slink||Topological surface}}). A ''differentiable surface'' is a surfaces that is a [[Differentiable manifold|differentiable manifold]] (see {{slink||Differentiable surface}}). Every differentiable surface is a topological surface, but the converse is false.

A "surface" is often implicitly supposed to be contained in a [[Euclidean space]] of dimension 3, typically {{math|'''R'''<sup>3</sup>}}. A surface that is contained in a [[Projective space|projective space]] is called a projective surface (see {{slink||Projective surface}}). A surface that is not supposed to be included in another space is called an ''abstract surface''.

==Examples==
* The [[Graph of a function|graph]] of a [[Continuous function|continuous function]] of two variables, defined over a [[Connected space|connected]] open subset of {{math|'''R'''<sup>2</sup>}} is a ''topological surface''. If the function is [[Differentiable function|differentiable]], the graph is a ''differentiable surface''.
* A [[Plane (geometry)|plane]] is both an [[Algebraic surface|algebraic surface]] and a differentiable surface. It is also a [[Ruled surface|ruled surface]] and a [[Surface of revolution|surface of revolution]].
* A circular cylinder (that is, the [[Locus (mathematics)|locus]] of a line crossing a circle and parallel to a given direction) is an algebraic surface and a differentiable surface.
* A [[Conical surface|circular cone]] (locus of a line crossing a circle, and passing through a fixed point, the ''apex'', which is outside the plane of the circle) is an algebraic surface which is not a differentiable surface. If one removes the apex, the remainder of the cone is the union of two differentiable surfaces. 
* The surface of a [[Polyhedron|polyhedron]] is a topological surface, which is neither a differentiable surface nor an algebraic surface.
* A hyperbolic paraboloid (the graph of the function {{math|1=''z'' = ''xy''}}) is a differentiable surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used in architecture.
* A two-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differentiable surfaces.

==Parametric surface==
{{main|Parametric surface}}
A '''parametric surface''' is the image of an open subset of the [[Euclidean plane]] (typically <math>\mathbb R^2</math>) by a [[Continuous function|continuous function]], in a [[Topological space|topological space]], generally a [[Euclidean space]] of dimension at least three. Usually the function is supposed to be continuously differentiable, and this will be always the case in this article.

Specifically, a parametric surface in <math>\mathbb R^3</math> is given by three functions of two variables {{mvar|u}} and {{mvar|v}}, called ''parameters''
:<math>\begin{align}
x&=f_1(u,v),\\[4pt]
y&=f_2(u,v),\\[4pt]
z&=f_3(u,v)\,.
\end{align}</math>

As the image of such a function may be a [[Curve|curve]] (for example, if the three functions are constant with respect to {{mvar|v}}), a further condition is required, generally that, for [[Almost all|almost all]] values of the parameters, the Jacobian matrix 
:<math>
\begin{bmatrix}
\dfrac{\partial f_1}{\partial u} & \dfrac{\partial f_1}{\partial v}\\[6pt]
\dfrac{\partial f_2}{\partial u} & \dfrac{\partial f_2}{\partial v}\\[6pt]
\dfrac{\partial f_3}{\partial u} & \dfrac{\partial f_3}{\partial v}
\end{bmatrix}
</math>
has rank two. Here "almost all" means that the values of the parameters where the rank is two contain a dense open subset of the range of the parametrization. For surfaces in a space of higher dimension, the condition is the same, except for the number of columns of the Jacobian matrix.

===Tangent plane and normal vector===
A point {{mvar|p}} where the above Jacobian matrix has rank two is called ''regular'', or, more properly, the parametrization is called ''regular'' at {{mvar|p}}.

The ''tangent plane'' at a regular point {{mvar|p}} is the unique plane passing through {{mvar|p}} and having a direction parallel to the two row vectors of the Jacobian matrix. The tangent plane is an affine concept, because its definition is independent of the choice of a [[Metric (mathematics)|metric]]. In other words, any [[Affine transformation|affine transformation]] maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.

The ''normal line'' at a point of a surface is the unique line passing through the point and perpendicular to the tangent plane; the ''normal vector'' is a vector which is parallel to the normal.

For other [[Differential invariant|differential invariant]]s of surfaces, in the neighborhood of a point, see [[Differential geometry of surfaces]].

===Irregular point and singular point===
A point of a parametric surface which is not regular is '''irregular'''. There are several kinds of irregular points.

It may occur that an irregular point becomes regular, if one changes the parametrization. This is the case of the poles in the parametrization of the [[Unit sphere|unit sphere]] by [[Euler angles]]: it suffices to permute the role of the different coordinate axes for changing the poles.

On the other hand, consider the circular cone of parametric equation
:<math>\begin{align}
x&= t\cos(u)\\
y&=t\sin(u)\\
z&=t\,.
\end{align}</math>
The apex of the cone is the origin {{math|(0, 0, 0)}}, and is obtained for {{math|1=''t'' = 0}}. It is an irregular point that remains irregular, whichever parametrization is chosen (otherwise, there would exist a unique tangent plane). Such an irregular point, where the tangent plane is undefined, is said '''singular'''.

There is another kind of singular points. There are the '''self-crossing points''', that is the points where the surface crosses itself. In other words, these are the points which are obtained for (at least) two different values of the parameters.

===Graph of a bivariate function===

Let {{math|1=''z'' = ''f''(''x'', ''y'')}} be a function of two real variables. This is a parametric surface, parametrized as 
:<math>\begin{align}
x&= t\\
y&=u\\
z&=f(t,u)\,.
\end{align}</math>
Every point of this surface is regular, as the two first columns of the Jacobian matrix form the [[Identity matrix|identity matrix]] of rank two.

===Rational surface===
{{main|Rational surface}}
A '''rational surface''' is a surface that may be parametrized by rational functions of two variables. That is, if {{math|''f<sub>i</sub>''(''t'', ''u'')}} are, for {{math|1=''i'' = 0, 1, 2, 3}}, [[Polynomial|polynomial]]s in two indeterminates, then the parametric surface, defined by 
:<math>\begin{align}
x&= \frac{f_1(t,u)}{f_0(t,u)},\\[6pt]
y&=\frac{f_2(t,u)}{f_0(t,u)},\\[6pt]
z&=\frac{f_3(t,u)}{f_0(t,u)}\,, 
\end{align}</math>
is a rational surface.

A rational surface is an [[Algebraic surface|algebraic surface]], but most algebraic surfaces are not rational.

==Implicit surface==
{{main|Implicit surface}}
An implicit surface in a [[Euclidean space]] (or, more generally, in an [[Affine space|affine space]]) of dimension 3 is the set of the common zeros of a [[Differentiable function|differentiable function]] of three variables
:<math>f(x, y, z)=0.</math>

Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the [[Implicit function theorem|implicit function theorem]]: if {{math|1=''f''(''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) = 0}}, and the partial derivative in {{mvar|z}} of {{mvar|f}} is not zero at {{math|(''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>)}}, then there exists a differentiable function {{math|''φ''(''x'', ''y'')}} such that
:<math>f(x,y,\varphi(x,y))=0</math>
in a [[Neighbourhood (mathematics)|neighbourhood]] of {{math|(''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>)}}. In other words, the implicit surface is the [[Graph of a function|graph of a function]] near a point of the surface where the partial derivative in {{mvar|z}} is nonzero. An implicit surface has thus, locally, a parametric representation, except at the points of the surface where the three partial derivatives are zero.

===Regular points and tangent plane===
A point of the surface where at least one partial derivative of {{mvar|f}} is nonzero is called '''regular'''. At such a point <math>(x_0, y_0, z_0)</math>, the tangent plane and the direction of the normal are well defined, and may be deduced, with the implicit function theorem from the definition given above, in {{slink||Tangent plane and normal vector}}. The direction of the normal is the [[Gradient|gradient]], that is the vector
:<math>\left[\frac{\partial f}{\partial x}(x_0, y_0, z_0), \frac{\partial f}{\partial y}(x_0, y_0, z_0), \frac{\partial f}{\partial z}(x_0, y_0, z_0)\right].</math>
The tangent plane is defined by its implicit equation
:<math>\frac{\partial f}{\partial x}(x_0, y_0, z_0)(x-x_0) + \frac{\partial f}{\partial y}(x_0, y_0, z_0) (y-y_0)+ \frac{\partial f}{\partial z}(x_0, y_0, z_0)(z-z_0) = 0.</math>

===Singular point===
A '''singular point''' of an implicit surface (in <math>\mathbb R^3</math>) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a system of four equations in three indeterminates. As most such systems have no solution, many surfaces do not have any singular point. A surface with no singular point is called ''regular'' or ''non-singular''.

The study of surfaces near their singular points and the classification of the singular points is [[Singularity theory|singularity theory]]. A singular point is [[Isolated singularity|isolated]] if there is no other singular point in a neighborhood of it. Otherwise, the singular points may form a curve. This is in particular the case for self-crossing surfaces.

==Algebraic surface==
{{main|Algebraic surface}}
Originally, an algebraic surface was a surface which may be defined by an implicit equation
:<math>f(x,y,z)=0,</math>
where {{math|''f''}} is a polynomial in three [[Indeterminate (variable)|indeterminate]]s, with real coefficients.

The concept has been extended in several directions, by defining surfaces over arbitrary [[Field (mathematics)|field]]s, and by considering surfaces in spaces of arbitrary dimension or in [[Projective space|projective space]]s. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.

===Surfaces over arbitrary fields===
Polynomials with coefficients in any [[Field (mathematics)|field]] are accepted for defining an algebraic surface. 
However, the field of coefficients of a polynomial is not well defined, as, for example, a polynomial with [[Rational number|rational]] coefficients may also be considered as a polynomial with [[Real number|real]] or [[Complex number|complex]] coefficients. Therefore, the concept of ''point'' of the surface has been generalized in the following way.<ref>{{Citation | last1=Weil | first1=André |  title=Foundations of Algebraic Geometry | url=https://books.google.com/books?id=ML7u26rkEkIC | publisher=American Mathematical Society | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications |volume=29  |mr=0023093 | year=1946| isbn=9780821874622 |pages = 1–363}}{{page needed|date = February 2022}}</ref>{{page needed|date = February 2022}}

Given a polynomial {{math|''f''(''x'', ''y'', ''z'')}}, let {{math|''k''}} be the smallest field containing the coefficients, and {{math|''K''}} be an algebraically closed extension of {{math|''k''}}, of infinite [[Transcendence degree|transcendence degree]].<ref>The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of [[Generic point|generic point]].</ref> Then a ''point'' of the surface is an element of {{math|''K''<sup>3</sup>}} which is a solution of the equation
:<math>f(x,y,z)=0.</math>
If the polynomial has real coefficients, the field {{math|''K''}} is the complex field, and a point of the surface that belongs to <math>\mathbb{R}^3</math> (a usual point) is called a ''real point''. A point that belongs to {{math|''k''<sup>3</sup>}} is called ''rational over {{math|k}}'', or simply a ''rational point'', if {{math|''k''}} is the field of [[Rational number|rational number]]s.

===Projective surface===
A '''projective surface''' in a [[Projective space|projective space]] of dimension three is the set of points whose [[Homogeneous coordinates|homogeneous coordinates]] are zeros of a single [[Homogeneous polynomial|homogeneous polynomial]] in four variables. More generally, a projective surface is a subset of a projective space, which is a [[Projective variety|projective variety]] of [[Dimension of an algebraic variety|dimension]] two.

Projective surfaces are strongly related to affine surfaces (that is, ordinary algebraic surfaces). One passes from a projective surface to the corresponding affine surface by setting to one some coordinate or indeterminate of the defining polynomials (usually the last one). Conversely, one passes from an affine surface to its associated projective surface (called ''projective completion'') by [[Homogeneous polynomial#Homogenization|homogenizing]] the defining polynomial (in case of surfaces in a space of dimension three), or by homogenizing all polynomials of the defining ideal (for surfaces in a space of higher dimension).

===In higher dimensional spaces===
One cannot define the concept of an algebraic surface in a space of dimension higher than three without a general definition of an [[Algebraic variety|algebraic variety]] and of the [[Dimension of an algebraic variety|dimension of an algebraic variety]]. In fact, an algebraic surface is an ''algebraic variety of dimension two''.

More precisely, an algebraic surface in a space of dimension {{mvar|n}} is the set of the common zeros of at least {{math|''n'' – 2}} polynomials, but these polynomials must satisfy further conditions that may be not immediate to verify. Firstly, the polynomials must not define a variety or an [[Algebraic set|algebraic set]] of higher dimension, which is typically the case if one of the polynomials is in the [[Ideal (ring theory)|ideal]] generated by the others. Generally, {{math|''n'' – 2}} polynomials define an algebraic set of dimension two or higher. If the dimension is two, the algebraic set may have several [[Irreducible component|irreducible component]]s. If there is only one component the {{math|''n'' – 2}} polynomials define a surface, which is a [[Complete intersection|complete intersection]]. If there are several components, then one needs further polynomials for selecting a specific component.

Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces all algebraic sets whose irreducible components have the dimension two.

In the case of surfaces in a space of dimension three, every surface is a complete intersection, and a surface is defined by a single polynomial, which is [[Irreducible polynomial|irreducible]] or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.

==Topological surface==
{{main|Surface (topology)}}
In [[Topology|topology]], a surface is generally defined as a [[Manifold|manifold]] of dimension two. This means that a topological surface is a [[Topological space|topological space]] such that every point has a neighborhood that is [[Homeomorphism|homeomorphic]] to an open subset of a [[Euclidean plane]].

Every topological surface is homeomorphic to a polyhedral surface such that all [[Facet (geometry)|facets]] are [[Triangle|triangle]]s. The [[Combinatorics|combinatorial]] study of such arrangements of triangles (or, more generally, of higher-dimensional [[Simplex|simplex]]es) is the starting object of [[Algebraic topology|algebraic topology]]. This allows the characterization of the properties of surfaces in terms of purely algebraic [[Invariant (mathematics)|invariants]], such as the [[Genus (mathematics)|genus]] and homology groups.

The homeomorphism classes of surfaces have been completely described (see [[Surface (topology)]]).

==Differentiable surface==
{{excerpt|Differentiable surface}}

==Fractal surface==
{{excerpt|Fractal surface}}

==In computer graphics==
{{excerpt|Surface (computer graphics)}}

==See also==
* Area element, the area of a differential element of a surface
* Coordinate surfaces
* [[Hypersurface]]
* [[Perimeter]], a two-dimensional equivalent
* Polyhedral surface
* [[Shape]]
* [[Signed distance function]]
* Solid figure
* [[Surface area]]
* Surface patch
* [[Surface integral]]

==Notes==
{{reflist}}

==Sources==
<!-- added for excerpt -->
{{citation|last=Gauss|first=Carl Friedrich|title=General Investigations of Curved Surfaces of 1825 and 1827 |url= http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABR1255 |year=1902 |publisher=Princeton University Library}}

[[Category:Geometry]]
[[Category:Topology]]
[[Category:Surfaces| ]]

{{Sourceattribution|Surface (mathematics)}}