Surface (mathematics)

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In mathematics, a surface is a generalization of a plane which needs not be flat – that is, the curvature is not necessarily zero. This is analogous to a 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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Plane (geometry) Flat, two-dimensional surface

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

Curve object similar to a line but which is not required to be straight

In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that it may be curved.

Contents

The mathematical concept of surface is an idealization of what is meant by surface in common language, science, and computer graphics.

Science systematic enterprise that builds and organizes knowledge

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

Computer graphics graphics created using computers

Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with the help of specialized graphical hardware and software. It is a vast and recently developed area of computer science. The phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes erroneously referred to as computer-generated imagery (CGI).

A sphere is the surface of a solid ball, here having radius r. Sphere and Ball.png
A sphere is the surface of a solid ball, here having radius r.

Definitions

Often, a surface is defined by equations that are satisfied by the coordinates of its points. This is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface. [1] If the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation

In mathematics, an equation is a statement that asserts the equality of two expressions. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English any equality is an equation.

Graph of a function the set of the couples (x,f(x)), with x is in the domain of the function f

In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f ), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and form thus a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function.

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

A surface may also be defined as the image, in some space of dimension at least 3, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a 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 longitude u and latitude v by

Image (mathematics) The set of all values of a function

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

Dimension Maximum number of independent directions within a mathematical space

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

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 2π). For the remaining two points (the north and south poles), one has cos v = 0, and the longitude 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: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is 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 singularities, such as the vertex of a conical surface or points where a surface crosses itself.

Modulo operation

In computing, the modulo operation finds the remainder after division of one number by another.

North Pole Northern point where the Earths axis of rotation intersects its surface

The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is defined as the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface.

South Pole Southern point where the Earths axis of rotation intersects its surface

The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of Earth and lies on the opposite side of Earth from the North Pole.

In classical geometry, a surface is generally defined as a locus of a point or a line. For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line. A 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 of lines.

Locus (mathematics) set of points whose location satisfies or is determined by one or more specified conditions

In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions.

Sphere round geometrical and circular object in three-dimensional space; special case of spheroid

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

Conical surface

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. Each of those lines is called a generatrix of the surface.

Terminology

In this article, several kinds of surfaces are considered and compared. An unambiguous terminology is thus necessary to distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two (the surfaces considered in Surface (topology)). We call differential surfaces the surfaces that are differentiable manifolds (the surfaces considered in Surface (differential geometry)). Every differential surface is a topological surface, but the converse is false.

For simplicity, unless otherwise stated, "surface" will mean a surface in the Euclidean space of dimension 3 or in R3. A surface that is not supposed to be included in another space is called an abstract surface.

Examples

Parametric surface

A 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

A 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 , or simply normal at a point of a surface is the unique line passing through the point and perpendicular to the tangent plane. A 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

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

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

Let 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

A 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

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

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

A 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

A 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

Originally, 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

Polynomials 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]

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

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 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

One cannot define the concept of 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 algebraic sets all of 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.

Abstract algebraic surface

Rational surfaces are algebraic surfaces

Topological surface

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 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)).

Differential surface

Fractal surface

In computer graphics

See also

Notes

  1. Here implicit does not refer to a property of the surface, which may be defined by other means, but it refers to how it is defined. Thus this term is an abbreviation of "surface defined by an implicit equation".
  2. Weil, André (1946), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, MR   0023093
  3. The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of generic point.

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Tangent straight line touching a point in a curve

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f ) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

Curvature physical quantity

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space – in a way that relates to the radius of curvature of circles that touch the object – and intrinsic curvature, which is defined in terms of the lengths of curves within a Riemannian manifold.

Vector field assignment of a vector to each point in a subset of Euclidean space

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Normal (geometry) in geometry, an object that is perpendicular to a given object

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

Algebraic curve Curve defined as zeros of polynomials

In mathematics, a affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve can be restricted to an affine algebraic plane curve by replacing by one some indeterminate of the defining homogeneous polynomial. As these two operations are each inverse to the other, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally.

Parametric equation parametric system of coordinates

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

Affine connection Construct allowing differentiation of tangent vector fields of manifolds

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves. Plane curves also include the Jordan curves and the graphs of continuous functions.

Singular point of an algebraic variety

In the mathematical field of algebraic geometry, a singular point of an algebraic varietyV is a point P that is 'special', in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

Implicit surface surface in Euclidean space

In mathematics an implicit surface is a surface in Euclidean space defined by an equation

Critical point (mathematics) in calculus, the points of an equation where the derivative is zero

Critical point is a wide term used in a lot of branches of mathematics, but is always connected to the derivative of a function or mapping.

Dual curve

In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of C, given in line coordinates, is known as the tangential equation of C.

Cusp (singularity) Point on a curve where motion must move backwards

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

Differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, 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 the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.