Surface (mathematics)

There are several more precise definitions, depending on the context and the mathematical tools that are used for the study.

For example, the surface of the Earth resembles (ideally) a sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).

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 ensure that the image is not a curve).

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.

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 "surface" is often implicitly supposed to be contained in a Euclidean space of dimension 3, typically R3.

A parametric surface is the image of an open subset of the Euclidean plane (typically

Usually the function is supposed to be continuously differentiable, and this will be always the case in this article.

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.

A point of the surface where at least one partial derivative of f is nonzero is called regular.

) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero.

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.

[2][page needed] 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.

Generally, n – 2 polynomials define an algebraic set of dimension two or higher.

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.

One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss,[4] who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

[5] Many natural phenomena exhibit some form of statistical self-similarity that can be modeled by fractal surfaces.

[6] Moreover, variations in surface texture provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects.

[7] The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoit Mandelbrot.