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{{short description|Idealisasi matematis permukaan benda}}
[[Gambar:Sphere and Ball.png|right|thumb|Sebuah [[bola]] adalah permukaan [[bola (matematika)|bola]] padat, ini memiliki [[jari-jari]] ''r'']]
Dalam [[matematika]], '''permukaan''' adalah [[model matematika]] dari konsep umum [[permukaan]]. Ini merupakan sebuah generalisasi dari [[bidang (matematika)|bidang]], namun tampak tidak seperti bidang kemungkinan termasuk [[lengkungan]] dengan [[kurva]] yang menggeneralisasi [[garis lurus]].
Ada beberapa definisi yang tepat, tergantung pada konteks dan alat matematika digunakan untuk penelitian. Permukaan matematika paling sederhana adalah bidang dan [[bola]] di [[ruang-3 Euklidean]]. Definisi yang tepat dari permukaan mungkin tergantung pada konteksnya. Biasanya, dalam [[geometri aljabar]], sebuah permukaan dapat bersilangan dengan sendiri (dan mungkin memiliki [[titik tunggal dari variasi aljabar|singularitas]] lainnya), sedangkan, dalam [[topologi]] dan [[geometri diferensial]] kemungkinan tidak.
Permukaan adalah [[ruang topologi]] dari [[dimensi]] dua; ini berarti bahwa titik bergerak pada permukaan yang mungkin bergerak dalam dua arah (memiliki dua [[derajat kebebasan]]). Dengan kata lain, di sekitar hampir setiap titik, terdapat ''[[patok koordinat]]'' yang dimana [[sistem koordinat]] dua dimensi ditentukan. Misalnya, permukaan Bumi menyerupai (idealnya) [[bola]] dua dimensi, dan [[lintang]] dan [[bujur]] memberikan koordinat dua dimensi di atasnya (kecuali di kutub dan sepanjang [[meridian ke-180]]).
==Definisi==
Seringkali, suatu permukaan ditentukan oleh [[persamaan]] yang dipenuhi oleh [[koordinat]] titik-titiknya. Inilah adalah kasus [[grafik fungsi|grafik]] dari [[fungsi kontinu]] dari dua variabel. Himpunan [[nol fungsi]] dari tiga variabel adalah permukaan yang disebut [[permukaan implisit]].<ref>Di sini "implisit" tidak mengacu pada sifat permukaan yang dapat didefinisikan dengan cara lain, melainkan bagaimana hal itu didefinisikan. Jadi istilah ini adalah singkatan dari "permukaan didefinisikan oleh [[persamaan implisit]]".</ref> Jika fungsi tiga variabel yang menentukan adalah [[polinomial]], permukaannya adalah [[permukaan aljabar]]. Misalnya, [[unit bola]] adalah permukaan aljabar, seperti yang didefinisikan oleh [[persamaan implisit]]
:<math>x^2+y^2+z^2 -1= 0.</math>
Permukaan juga dapat didefinisikan sebagai [[Gambar (matematika)|gambar]], suatu ruang dengan [[dimensi]] setidaknya 3, dari [[fungsi kontinu]] dari dua variabel dari beberapa kondisi lebih lanjut diperlukan untuk memastikan bahwa gambar tersebut bukan [[kurva]]. Dalam hal ini, ini menyatakan bahwa memiliki [[permukaan parametrik]], "parametriks" oleh dua variabel ini yang disebut "parameter". Misalnya, unit bola dapat diparametrikan oleh [[sudut Euler]], juga disebut [[bujur]] <math>u</math> dan [[lintang]] <math>v</math> oleh
:<math>\begin{align}
x&= \cos(u)\cos(v)\\
y&=\sin(u)\cos(v)\\
z&=\sin(v)\,.
\end{align}</math>
Persamaan parametrik permukaan seringkali tidak beraturan pada beberapa titik. Misalnya, semua kecuali dua titik dari unit bola adalah bayangan dengan parameterisasi di atas, tepat sepasang sudut Euler ([[operasi modulo|modulo]] <math>2 \pi</math>). Untuk dua titik yang tersisa pada kutub ([[Kutub utara|utara]] dan [[Kutub selatan|selatan]]) ada <math>\cos(v) = 0</math>, dan bujur <math>u</math> dapat memiliki nilai berapa pun. Juga, terdapat permukaan yang tidak bisa ada parameterisasi tunggal yang menutupi seluruh permukaan. Oleh karena itu, satu-satunya yang sering menganggap permukaan diparametrikan oleh beberapa persamaan parametrik, yang bayangannya menutupi permukaan. Ini diformalkan oleh konsep [[manifold]]: dalam konteks manifold, biasanya dalam [[topologi]] dan [[geometri diferensial]], sebuah permukaan adalah manifold berdimensi dua; ini berarti bahwa permukaan adalah [[ruang topologi]] sehingga setiap titik memiliki [[lingkungan]] yang [[homeomorfisme|homeomorfik]] ke [[himpunan bagian terbuka]] dari [[bidang Euklides]] (lihat [[Permukaan (topologi)]] dan [[Permukaan (geometri diferensial)]]). Hal ini memungkinkan pendefinisian permukaan dalam ruang berdimensi lebih tinggi dari tiga, dan bahkan "permukaan abstrak" yang tidak terkandung dalam ruang lain mana pun. Di sisi lain, ini mengecualikan permukaan yang memiliki [[teori singularitas|singularitas]], seperti puncak [[permukaan berbentuk kerucut]] atau titik dimana permukaan melintasi dirinya sendiri.
Dalam [[geometri klasik]], permukaan umumnya didefinisikan sebagai [[lokus (matematika)|lokus]] titik atau garis. Misalnya, [[bola (geometri)|bola]] adalah tempat kedudukan suatu titik yang berada pada jarak tertentu dari suatu titik tetap yang disebut pusat; sebuah [[permukaan kerucut]] adalah tempat kedudukan garis yang melewati titik tetap dan melintasi [[kurva]]; sebuah [[permukaan revolusi]] adalah lokus kurva yang berputar pada sekitar garis. Sebuah [[permukaan beraturan]] adalah lokus garis bergerak yang memenuhi beberapa kendala; dalam terminologi modern, permukaan beraturan adalah permukaan yang merupakan [[penyatuan (teori himpunan)|penyatuan]] garis.
==Terminologi==
Ada beberapa jenis permukaan yang dipertimbangkan dalam matematika. Oleh karena itu, terminologi yang tidak ambigu diperlukan untuk membedakannya bila diperlukan. Sebuah ''[[permukaan topologi]]'' adalah permukaan yang merupakan [[manifold]] berdimensi dua (lihat {{slink||Permukaan Tologi}}). Sebuah ''[[permukaan diferensial]]'' adalah permukaan yang merupakan [[manifold diferensial]] (lihat {{slink||Permukaan diferensial}}). Setiap permukaan diferensial adalah permukaan topologi, tetapi kebalikannya salah.
Sebuah "permukaan" sering secara implisit dianggap terkandung dalam [[ruang Euklides]] berdimensi 3, biasanya <math>\mathbf R^3</math>. Permukaan yang dimuat dalam [[ruang proyektif]] disebut [[permukaan proyektif]] (lihat {{slink||Permukaan proyektif}}). Permukaan yang tidak seharusnya dimasukkan ke dalam ruang lain disebut ''permukaan abstrak''.
==Examples==
* The [[graph of a function|graph]] of a [[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]] and a differentiable surface. It is also a [[ruled surface]] and a [[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]] 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]], in a [[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)\\
y&=f_2(u,v)\\
z&=f_3(u,v)\,.
\end{align}</math>
As the image of such a function may be a [[curve]] (for example, if the three functions are constant with respect to {{mvar|v}}), a further condition is required, generally that, for [[almost all]] values of the parameters, the [[Jacobian matrix]]
:<math>
\begin{bmatrix}
\dfrac{\partial f_1}{\partial u} & \dfrac{\partial f_1}{\partial v}\\
\dfrac{\partial f_2}{\partial u} & \dfrac{\partial f_2}{\partial v}\\
\dfrac{\partial f_3}{\partial u} & \dfrac{\partial f_3}{\partial v}\\
\end{bmatrix}
</math>
has [[rank of a matrix|rank]] two. Here "almost all" means that the values of the parameters where the rank is two contain a [[dense subset|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 vector]]s of the Jacobian matrix. The tangent plane is an [[affine property|affine concept]], because its definition is independent of the choice of a [[metric (mathematics)|metric]]. In other words, any [[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]]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]] 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]] 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]]s in two indeterminates, then the parametric surface, defined by
:<math>\begin{align}
x&= \frac{f_1(t,u)}{f_0(t,u)}\\
y&=\frac{f_2(t,u)}{f_0(t,u)}\\
z&=\frac{f_3(t,u)}{f_0(t,u)}\,,
\end{align}</math>
is a rational surface.
A rational surface is an [[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]]) of dimension 3 is the set of the common zeros of a [[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]]: 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]] 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]], 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
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 [[simultaneous equations|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]]. 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]]s. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.
===
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 | author1-link=André Weil | 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]].<ref>The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of [[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]]s.
===
A '''projective surface''' in a [[projective space]] of dimension three is the set of points whose [[homogeneous coordinates]] are zeros of a single [[homogeneous polynomial]] in four variables. More generally, a projective surface is a subset of a projective space, which is a [[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]] and of the [[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]] 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]]s. If there is only one component the {{math|''n'' – 2}} polynomials define a surface, which is a [[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.
==
{{main|Surface (topology)}}
In [[topology]], a surface is generally defined as a [[manifold]] of dimension two. This means that a topological surface is a [[topological space]] such that every point has a [[neighborhood (mathematics)|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]]s. The [[combinatorics|combinatorial]] study of such arrangements of triangles (or, more generally, of higher-dimensional [[simplex]]es) is the starting object of [[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 group]]s.
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}}
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