# algebraic surface

﻿
algebraic surface

in three-dimensional space, a surface the equation of which is f(xyz) = 0, with f(xyz) a polynomial in x, y, z. The order of the surface is the degree of the polynomial equation. If the surface is of the first order, it is a plane. If the surface is of order two, it is called a quadric surface. By rotating the surface, its equation can be put in the form

Ax2 + By2 + Cz2 + Dx + Ey + Fz = G.

If A, B, C are all not zero, the equation can generally be simplified to the form
ax2 + by2 + cz2 = 1.
This surface is called an ellipsoid if a, b, and c are positive. If one of the coefficients is negative, the surface is a hyperboloid of one sheet; if two of the coefficients are negative, the surface is a hyperboloid of two sheets. A hyperboloid of one sheet has a saddle point (a point on a curved surface shaped like a saddle at which the curvatures in two mutually perpendicular planes are of opposite signs, just like a saddle is curved up in one direction and down in another).

If A, B, C are possibly zero, then cylinders, cones, planes, and elliptic or hyperbolic paraboloids may be produced. Examples of the latter are z = x2 + y2 and z = x2 −  y2, respectively. Through every point of a quadric pass two straight lines lying on the surface. A cubic surface is one of order three. It has the property that 27 lines lie on it, each one meeting 10 others. In general, a surface of order four or more contains no straight lines.

* * *

Universalium. 2010.

### Look at other dictionaries:

• Algebraic surface — In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two (as a complex manifold, when it is non singular)… …   Wikipedia

• algebraic surface — noun : a surface expressible analytically through an algebraic equation connecting its Cartesian coordinates …   Useful english dictionary

• Surface — This article discusses surfaces from the point of view of topology. For other uses, see Differential geometry of surfaces, algebraic surface, and Surface (disambiguation). An open surface with X , Y , and Z contours shown. In mathematics,… …   Wikipedia

• Surface of general type — In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2.These are all algebraic, and in some sense most surfaces are in this class. ClassificationGieseker showed that there is a coarse moduli scheme for… …   Wikipedia

• Algebraic geometry — This Togliatti surface is an algebraic surface of degree five. Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It… …   Wikipedia

• Algebraic space — In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory.DefinitionAn algebraic space X comprises a schemeOne can always assume that U is an affine… …   Wikipedia

• Algebraic statistics — is a fairly recent field of statistics which utilizes the tools of algebraic geometry and commutative algebra in order to study problems related to discrete random variables with finite state spaces. Such problems include parameter estimation,… …   Wikipedia

• Algebraic curve — In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with… …   Wikipedia

• Algebraic geometry and analytic geometry — In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally… …   Wikipedia

• Algebraic function — In mathematics, an algebraic function is informally a function which satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x is a solution y for an equation: a n(x)y^n+a… …   Wikipedia