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Inmathematics, asurface is amathematical model of the common concept of asurface. It is a generalization of aplane, but, unlike a plane, it may becurved; this is analogous to acurve generalizing astraight 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 andspheres in theEuclidean 3-space. The exact definition of a surface may depend on the context. Typically, inalgebraic geometry, a surface may cross itself (and may have othersingularities), while, intopology anddifferential geometry, it may not.

A surface is atopological space ofdimension two; this means that a moving point on a surface may move in two directions (it has twodegrees of freedom). In other words, around almost every point, there is acoordinate patch on which atwo-dimensionalcoordinate system is defined. For example, the surface of the Earth resembles (ideally) a sphere, andlatitude andlongitude provide two-dimensional coordinates on it (except at the poles and along the180th meridian).

Often, a surface is defined byequations that are satisfied by thecoordinates of its points. This is the case of thegraph of acontinuous function of two variables. The set of thezeros of a function of three variables is a surface, which is called animplicit surface.^[1] If the defining three-variate function is apolynomial, the surface is analgebraic surface. For example, theunit sphere is an algebraic surface, as it may be defined by theimplicit equation

${\displaystyle x^2+y^2+z^2-1=0.}$

A surface may also be defined as theimage, in some space ofdimension at least 3, of acontinuous function of two variables (some further conditions are required to ensure that the image is not acurve). In this case, one says that one has aparametric surface, which isparametrized by these two variables, calledparameters. For example, the unit sphere may be parametrized by theEuler angles, also calledlongitudeu andlatitudev by

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 (modulo2π). For the remaining two points (thenorth andsouth poles), one hascosv = 0, and the longitudeu 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 ofmanifold: in the context of manifolds, typically intopology anddifferential geometry, a surface is a manifold of dimension two; this means that a surface is atopological space such that every point has aneighborhood which ishomeomorphic to anopen subset of theEuclidean plane (seeSurface (topology) andSurface (differential geometry)). This allows defining surfaces in spaces of dimension higher than three, and evenabstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that havesingularities, such as the vertex of aconical surface or points where a surface crosses itself.

Inclassical geometry, a surface is generally defined as alocus of a point or a line. For example, asphere is the locus of a point which is at a given distance of a fixed point, called the center; aconical surface is the locus of a line passing through a fixed point and crossing acurve; asurface of revolution is the locus of a curve rotating around a line. Aruled surface is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is aunion of lines.

There are several kinds of surfaces that are considered in mathematics. An unambiguous terminology is thus necessary to distinguish them when needed. Atopological surface is a surface that is amanifold of dimension two (see§ Topological surface). Adifferentiable surface is a surfaces that is adifferentiable manifold (see§ Differentiable surface). Every differentiable surface is a topological surface, but the converse is false.

A "surface" is often implicitly supposed to be contained in aEuclidean space of dimension 3, typicallyR³. A surface that is contained in aprojective space is called aprojective surface (see§ Projective surface). A surface that is not supposed to be included in another space is called anabstract surface.

• Thegraph of acontinuous function of two variables, defined over aconnectedopen subset ofR² is atopological surface. If the function isdifferentiable, the graph is adifferentiable surface.
• Aplane is both analgebraic surface and a differentiable surface. It is also aruled surface and asurface of revolution.
• Acircular cylinder (that is, thelocus of a line crossing a circle and parallel to a given direction) is an algebraic surface and a differentiable surface.
• Acircular cone (locus of a line crossing a circle, and passing through a fixed point, theapex, 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 apolyhedron is a topological surface, which is neither a differentiable surface nor an algebraic surface.
• Ahyperbolic paraboloid (the graph of the functionz =xy) is a differentiable surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used inarchitecture.
• Atwo-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differentiable surfaces.

Aparametric surface is the image of an open subset of theEuclidean plane (typically${\displaystyle {\mathbb{R}}^2}$) by acontinuous function, in atopological space, generally aEuclidean space of dimension at least three. Usually the function is supposed to becontinuously differentiable, and this will be always the case in this article.

Specifically, a parametric surface in${\displaystyle {\mathbb{R}}^3}$ is given by three functions of two variablesu andv, calledparameters

As the image of such a function may be acurve (for example, if the three functions are constant with respect tov), a further condition is required, generally that, foralmost all values of the parameters, theJacobian matrix

hasrank two. Here "almost all" means that the values of the parameters where the rank is two contain adenseopen 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.

A pointp where the above Jacobian matrix has rank two is calledregular, or, more properly, the parametrization is calledregular atp.

Thetangent plane at a regular pointp is the unique plane passing throughp and having a direction parallel to the tworow vectors of the Jacobian matrix. The tangent plane is anaffine concept, because its definition is independent of the choice of ametric. In other words, anyaffine 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.

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

For otherdifferential invariants of surfaces, in the neighborhood of a point, seeDifferential geometry of surfaces.

A point of a parametric surface which is not regular isirregular. 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 theunit sphere byEuler angles: it suffices to permute the role of the differentcoordinate axes for changing the poles.

On the other hand, consider thecircular cone of parametric equation

The apex of the cone is the origin(0, 0, 0), and is obtained fort = 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 saidsingular.

There is another kind of singular points. There are theself-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.

Letz =f(x,y) be a function of two real variables, abivariate function. This is a parametric surface, parametrized as

Every point of this surface isregular, as the two first columns of the Jacobian matrix form theidentity matrix of rank two.

Arational surface is a surface that may be parametrized byrational functions of two variables. That is, iff_i(t,u) are, fori = 0, 1, 2, 3,polynomials in two indeterminates, then the parametric surface, defined by

is a rational surface.

A rational surface is analgebraic surface, but most algebraic surfaces are not rational.

An implicit surface in aEuclidean space (or, more generally, in anaffine space) of dimension 3 is the set of the common zeros of adifferentiable function of three variables

${\displaystyle f(x,y,z)=0.}$

Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by theimplicit function theorem: iff(x₀,y₀,z₀) = 0, and the partial derivative inz off is not zero at(x₀,y₀,z₀), then there exists a differentiable functionφ(x,y) such that

${\displaystyle f(x,y,\phi (x,y))=0}$

in aneighbourhood of(x₀,y₀,z₀). In other words, the implicit surface is thegraph of a function near a point of the surface where the partial derivative inz 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.

A point of the surface where at least one partial derivative off is nonzero is calledregular. At such a point${\displaystyle (x_0,y_0,z_0)}$, 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§ Tangent plane and normal vector. The direction of the normal is thegradient, that is the vector

${\displaystyle \left[\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(x_0,y_0,z_0),\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(x_0,y_0,z_0),\frac{\mathrm{\partial }f}{\mathrm{\partial }z}(x_0,y_0,z_0)\right].}$

The tangent plane is defined by its implicit equation

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}(x_0,y_0,z_0)(x-x_0)+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(x_0,y_0,z_0)(y-y_0)+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}(x_0,y_0,z_0)(z-z_0)=0.}$

Asingular point of an implicit surface (in${\displaystyle {\mathbb{R}}^3}$) 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 asystem 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 calledregular ornon-singular.

The study of surfaces near their singular points and the classification of the singular points issingularity theory. A singular point isisolated 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.

Originally, an algebraic surface was a surface which could be defined by an implicit equation

${\displaystyle f(x,y,z)=0,}$

wheref is a polynomial in threeindeterminates, with real coefficients.

The concept has been extended in several directions, by defining surfaces over arbitraryfields, and by considering surfaces in spaces of arbitrary dimension or inprojective spaces. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.

Polynomials with coefficients in anyfield are accepted for defining an algebraic surface. However, the field of coefficients of a polynomial is not well defined, as, for example, a polynomial withrational coefficients may also be considered as a polynomial withreal orcomplex coefficients. Therefore, the concept ofpoint of the surface has been generalized in the following way.^{[2][page needed]}

Given a polynomialf(x,y,z), letk be the smallest field containing the coefficients, andK be analgebraically closed extension ofk, of infinitetranscendence degree.^[3] Then apoint of the surface is an element ofK³ which is a solution of the equation

${\displaystyle f(x,y,z)=0.}$

If the polynomial has real coefficients, the fieldK is thecomplex field, and a point of the surface that belongs to${\displaystyle {\mathbb{R}}^3}$ (a usual point) is called areal point. A point that belongs tok³ is calledrational overk, or simply arational point, ifk is the field ofrational numbers.

Aprojective surface in aprojective space of dimension three is the set of points whosehomogeneous coordinates are zeros of a singlehomogeneous polynomial in four variables. More generally, a projective surface is a subset of a projective space, which is aprojective variety ofdimension 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 (calledprojective completion) byhomogenizing 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).

One cannot define the concept of an algebraic surface in a space of dimension higher than three without a general definition of analgebraic variety and of thedimension of an algebraic variety. In fact, an algebraic surface is analgebraic variety of dimension two.

More precisely, an algebraic surface in a space of dimensionn is the set of the common zeros of at leastn – 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 analgebraic set of higher dimension, which is typically the case if one of the polynomials is in theideal generated by the others. Generally,n – 2 polynomials define an algebraic set of dimension two or higher. If the dimension is two, the algebraic set may have severalirreducible components. If there is only one component then – 2 polynomials define a surface, which is acomplete 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 isirreducible or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.

Intopology, a surface is generally defined as amanifold of dimension two. This means that a topological surface is atopological space such that every point has aneighborhood that ishomeomorphic to anopen subset of aEuclidean plane.

Every topological surface is homeomorphic to apolyhedral surface such that allfacets aretriangles. Thecombinatorial study of such arrangements of triangles (or, more generally, of higher-dimensionalsimplexes) is the starting object ofalgebraic topology. This allows the characterization of the properties of surfaces in terms of purely algebraicinvariants, such as thegenus andhomology groups.

The homeomorphism classes of surfaces have been completely described (seeSurface (topology)).

This section is an excerpt fromDifferential geometry of surfaces.[edit]

Inmathematics, thedifferential geometry of surfaces deals with thedifferential geometry ofsmoothsurfaces^[a] with various additional structures, most often, aRiemannian metric.^[b]

Surfaces have been extensively studied from various perspectives:extrinsically, relating to theirembedding inEuclidean space andintrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is theGaussian curvature, first studied in depth byCarl Friedrich Gauss,^[4] who showed that curvature was an intrinsic property of a surface, independent of itsisometric embedding in Euclidean space.

Surfaces naturally arise asgraphs offunctions of a pair ofvariables, and sometimes appear in parametric form or asloci associated tospace curves. An important role in their study has been played byLie groups (in the spirit of theErlangen program), namely thesymmetry groups of theEuclidean plane, thesphere and thehyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry throughconnections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linearEuler–Lagrange equations in thecalculus of variations: although Euler developed the one variable equations to understandgeodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was tominimal surfaces, a concept that can only be defined in terms of an embedding.

This section is an excerpt fromFractal landscape.[edit]

Afractal landscape or fractal surface is generated using astochastic algorithm designed to producefractal behavior that mimics the appearance of naturalterrain. In other words, thesurface resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.^[5]

Many natural phenomena exhibit some form of statisticalself-similarity that can be modeled by fractal surfaces.^[6] Moreover, variations insurface texture provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects.^[7]The modeling of the Earth's rough surfaces viafractional Brownian motion was first proposed byBenoit Mandelbrot.^[8]

Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently applied to such landscapes that may affect thestationarity and even the overallfractal behavior of such a surface, in the interests of producing a more convincing landscape.

According toR. R. Shearer, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric,computer generated images and natural, man made art became blurred.^[9] The first use of a fractal-generated landscape in a film was in 1982 for the movieStar Trek II: The Wrath of Khan.Loren Carpenter refined the techniques of Mandelbrot to create an alien landscape.^[10]

This section is an excerpt fromComputer representation of surfaces.[edit]

In technical applications of3D computer graphics (CAx) such ascomputer-aided design andcomputer-aided manufacturing,surfaces are one way of representing objects. The other ways arewireframe (lines and curves) and solids.Point clouds are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.

• 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

• Gauss, Carl Friedrich (1902),General Investigations of Curved Surfaces of 1825 and 1827, Princeton University Library

• Geometry
• Topology
• Surfaces

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