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- Surface (mathematics)

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.

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

*x*^{2+y}^{2+z}^{2}-1=0*.*

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 and latitude by

*\begin{align}
x&*=*\cos(u)\cos(v)\\
y&*=*\sin(u)\cos(v)\\
z&*=*\sin(v).
\end{align}*

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). For the remaining two points (the north and south poles), one has, and the longitude 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 . 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 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) 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.

See main article: Parametric surface. A **parametric surface** is the image of an open subset of the Euclidean plane (typically

R^{2}

Specifically, a parametric surface in

R^{3}

*\begin{align}
x&*=*f*_{1(u,v)\\
y&=f}_{2(u,v)\\
z&=f}_{3(u,v).
\end{align}}

*\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}
*

A point where the above Jacobian matrix has rank two is called *regular*, or, more properly, the parametrization is called *regular* at .

The *tangent plane* at a regular point is the unique plane passing through 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

*\begin{align}
x&*=*t\cos(u)\\
y&*=*t\sin(u)\\
z&*=*t.
\end{align}*

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 be a function of two real variables. This is a parametric surface, parametrized as

*\begin{align}
x&*=*t\\
y&*=*u\\
z&*=*f(t,u).
\end{align}*

See main article: Rational surface. A **rational surface** is a surface that may be parametrized by rational functions of two variables. That is, if are, for, polynomials in two indeterminates, then the parametric surface, defined by

*\begin{align}
x&*=

f_{1(t,u)} | \\
y&= | |

f_{0(t,u)} |

f_{2(t,u)} | \\
z&= | |

f_{0(t,u)} |

f_{3(t,u)} | |

f_{0(t,u)} |

*,**\end{align}*

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

See main article: 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

*f(x,**y,**z)*=0*.*

Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the implicit function theorem: if, and the partial derivative in of is not zero at, then there exists a differentiable function such that

*f(x,y,\varphi(x,y))*=0

A point of the surface where at least one partial derivative of is nonzero is called **regular**. At such a point

*(x*_{0,}*y*_{0,}*z*_{0)}

\left[ | \partialf |

\partialx |

*(x*_{0,}*y*_{0,}*z*_{0),}

\partialf | |

\partialy |

*(x*_{0,}*y*_{0,}*z*_{0),}

\partialf | |

\partialz |

*(x*_{0,}*y*_{0,}*z*_{0)\right].}

\partialf | |

\partialx |

*(x*_{0,}*y*_{0,}*z*_{0)(x-x}_{0)}+

\partialf | |

\partialy |

*(x*_{0,}*y*_{0,}*z*_{0)}*(y*-*y*_{0)+}

\partialf | |

\partialz |

*(x*_{0,}*y*_{0,}*z*_{0)(z-z}_{0)}=0*.*

A **singular point** of an implicit surface (in

R^{3}

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.

See main article: Algebraic surface. Originally, an algebraic surface was a surface which may be defined by an implicit equation

*f(x,y,z)*=0*,*

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:

Given a polynomial, let be the smallest field containing the coefficients, and be an algebraically closed extension of, of infinite transcendence degree.^{[2]} Then a *point* of the surface is an element of which is a solution of the equation

*f(x,y,z)*=0*.*

R^{3}

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 is the set of the common zeros of at least 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, 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 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.

See main article: 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 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)).

See main article: Differentiable surface.

See main article: Fractal surface.

See main article: Surface (computer graphics).

- 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 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".
- The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of generic point.