Pengguna:Klasüo/bak pasir/khusus/3
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Dalam matematika, permukaan adalah model matematika dari konsep umum permukaan. Ini merupakan sebuah generalisasi dari 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 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
suntingSeringkali, suatu permukaan ditentukan oleh persamaan yang dipenuhi oleh koordinat titik-titiknya. Inilah adalah kasus grafik dari fungsi kontinu dari dua variabel. Himpunan nol fungsi dari tiga variabel adalah permukaan yang disebut permukaan implisit.[1] Jika fungsi tiga variabel yang menentukan adalah polinomial, permukaannya adalah permukaan aljabar. Misalnya, unit bola adalah permukaan aljabar, seperti yang didefinisikan oleh persamaan implisit
Permukaan juga dapat didefinisikan sebagai 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 dan lintang oleh
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 (modulo ). Untuk dua titik yang tersisa pada kutub (utara dan selatan) ada , dan bujur 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 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 singularitas, seperti puncak permukaan berbentuk kerucut atau titik dimana permukaan melintasi dirinya sendiri.
Dalam geometri klasik, permukaan umumnya didefinisikan sebagai lokus titik atau garis. Misalnya, 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 garis.
Terminologi
suntingAda 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 § Permukaan Tologi). Sebuah permukaan diferensial adalah permukaan yang merupakan manifold diferensial (lihat § Permukaan diferensial). Setiap permukaan diferensial adalah permukaan topologi, tetapi kebalikannya salah.
Sebuah "permukaan" sering secara implisit dianggap terkandung dalam ruang Euklides berdimensi 3, biasanya . Permukaan yang dimuat dalam ruang proyektif disebut permukaan proyektif (lihat § Permukaan proyektif). Permukaan yang tidak seharusnya dimasukkan ke dalam ruang lain disebut permukaan abstrak.
Examples
sunting- The graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differentiable surface.
- A 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 of a line crossing a circle and parallel to a given direction) is an algebraic surface and a differentiable surface.
- A 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 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
suntingA parametric surface is the image of an open subset of the Euclidean plane (typically ) 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 is given by three functions of two variables u and v, called parameters
As the image of such a function may be a curve (for example, if the three functions are constant with respect to v), a further condition is required, generally that, for almost all values of the parameters, the Jacobian matrix
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
suntingA point p where the above Jacobian matrix has rank two is called regular, or, more properly, the parametrization is called regular at p.
The tangent plane at a regular point p is the unique plane passing through 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. 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 invariants of surfaces, in the neighborhood of a point, see Differential geometry of surfaces.
Irregular point and singular point
suntingA 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
The apex of the cone is the origin (0, 0, 0), and is obtained for 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
suntingLet z = f(x, y) be a function of two real variables. This is a parametric surface, parametrized as
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
suntingA rational surface is a surface that may be parametrized by rational functions of two variables. That is, if fi(t, u) are, for i = 0, 1, 2, 3, polynomials in two indeterminates, then the parametric surface, defined by
is a rational surface.
A rational surface is an algebraic surface, but most algebraic surfaces are not rational.
Implicit surface
suntingAn 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
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 f(x0, y0, z0) = 0, and the partial derivative in z of f is not zero at (x0, y0, z0), then there exists a differentiable function φ(x, y) such that
in a neighbourhood of (x0, y0, z0). In other words, the implicit surface is the graph of a function near a point of the surface where the partial derivative in 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
suntingA point of the surface where at least one partial derivative of f is nonzero is called regular. At such a point , 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 the gradient, that is the vector
The tangent plane is defined by its implicit equation
Singular point
suntingA singular point of an implicit surface (in ) 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. A singular point is 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
suntingOriginally, an algebraic surface was a surface which may be defined by an implicit equation
where f is a polynomial in three indeterminates, with real coefficients.
The concept has been extended in several directions, by defining surfaces over arbitrary fields, and by considering surfaces in spaces of arbitrary dimension or in projective spaces. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.
Surfaces over arbitrary fields
suntingPolynomials with coefficients in any 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 coefficients may also be considered as a polynomial with real or complex coefficients. Therefore, the concept of point of the surface has been generalized in the following way.[2][halaman dibutuhkan]
Given a polynomial f(x, y, z), let k be the smallest field containing the coefficients, and K be an algebraically closed extension of k, of infinite transcendence degree.[3] Then a point of the surface is an element of K3 which is a solution of the equation
If the polynomial has real coefficients, the field K is the complex field, and a point of the surface that belongs to (a usual point) is called a real point. A point that belongs to k3 is called rational over k, or simply a rational point, if k is the field of rational numbers.
Projective surface
suntingA 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 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 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
suntingOne 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 n is the set of the common zeros of at least 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 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 several irreducible components. If there is only one component the 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 or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.
Topological surface
suntingIn 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 that is homeomorphic to an open subset of a Euclidean plane.
Every topological surface is homeomorphic to a polyhedral surface such that all facets are triangles. The combinatorial study of such arrangements of triangles (or, more generally, of higher-dimensional simplexes) is the starting object of algebraic topology. This allows the characterization of the properties of surfaces in terms of purely algebraic invariants, such as the genus and homology groups.
The homeomorphism classes of surfaces have been completely described (see Surface (topology)).
Differentiable surface
suntingFractal surface
suntingIn computer graphics
suntingSee also
sunting- 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
sunting- ^ 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".
- ^ Weil, André (1946), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, hlm. 1–363, ISBN 9780821874622, MR 0023093[halaman dibutuhkan]
- ^ The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of generic point.