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In mathematics, a **surface** is a generalization of a plane. Unlike a plane, it need not be flat – that is, its curvature need not be zero. This is analogous to a curve generalizing a straight line. There are many more-precise definitions, depending on the context and the mathematical tools used to analyze the surface.

- Definitions
- Terminology
- Examples
- Parametric surface
- Tangent plane and normal vector
- Irregular point and singular point
- Graph of a bivariate function
- Rational surface
- Implicit surface
- Regular points and tangent plane
- Singular point
- Algebraic surface
- Surfaces over arbitrary fields
- Projective surface
- In higher dimensional spaces
- Abstract algebraic surface
- Rational surfaces are algebraic surfaces
- Topological surface
- Differentiable surface
- Fractal surface
- In computer graphics
- See also
- Notes

The mathematical concept idealizes what is meant by * surface * in science, computer graphics, and common language.

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

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

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.

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.

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 differentiable surfaces the surfaces that are differentiable manifolds (the surfaces considered in Surface (differential geometry)). Every differentiable 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 **R**^{3}. A surface that is not supposed to be included in another space is called an **abstract surface**.

- The graph of a continuous function of two variables, defined over a connected open subset of
**R**^{2}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.

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.

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 * 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.

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.

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.

A **rational surface** is a surface that may be parametrized by rational functions of two variables. That is, if *f _{i}*(

is a rational surface.

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

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*(*x*_{0}, *y*_{0}, *z*_{0}) = 0, and the partial derivative in z of f is not zero at (*x*_{0}, *y*_{0}, *z*_{0}), then there exists a differentiable function *φ*(*x*, *y*) such that

in a neighbourhood of (*x*_{0}, *y*_{0}, *z*_{0}). 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.

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

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.

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.

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 *K*^{3} 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 *k*^{3} is called *rational over k*, or simply a *rational point*, if *k* is the field of rational numbers.

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).

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 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.

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)).

This section should include a summary of Differentiable surface.(May 2021) |

- Area element, the area of a differential element of a surface
- Coordinate surfaces
- Perimeter, a two-dimensional equivalent
- Polyhedral surface
- Shape
- Signed distance function
- Solid figure
- Surface area
- Surface integral

- ↑ 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".
- ↑ Weil, André (1946),
*Foundations of Algebraic Geometry*, American Mathematical Society Colloquium Publications,**29**, Providence, R.I.: American Mathematical Society, ISBN 9780821874622, MR 0023093 - ↑ The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of generic point.

In mathematics, an **equation** is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". 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.

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* 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.

A **vector space** is a set of objects called *vectors*, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector *axioms*. To specify that the scalars are real or complex numbers, the terms **real vector space** and **complex vector space** are often used.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In mathematics, a **curve** is an object similar to a line, but that does not have to be straight.

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the **normal line** to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In mathematics, an **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 of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

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.

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 immersed submanifold in a Riemannian manifold.

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.

In the mathematical field of algebraic geometry, a **singular point of an algebraic variety***V* 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**.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

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

**Critical point** is a wide term used in many branches of mathematics.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

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.

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.

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

In mathematics, an **implicit curve** is a plane curve defined by an implicit equation relating two coordinate variables, commonly *x* and *y*. For example, the unit circle is defined by the implicit equation . In general, every implicit curve is defined by an equation of the form

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.

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