Two-dimensional space

Edit links
Source: Wikipedia, the free encyclopedia.

A two-dimensional space is a

coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces
.

Euclidean space has parallel lines which extend infinitely while remaining equidistant. In non-Euclidean spaces, lines perpendicular to a traversal either converge or diverge.

The most basic example is the flat

parallel, meaning they never intersect
and stay at uniform distance from each-other.

Two-dimensional spaces can also be

three-dimensional Euclidean space or some other ambient space, and inherit their structure from it; for example, ruled surfaces such as the cylinder and cone contain a straight line through each point, and minimal surfaces locally minimize their area, as is done physically by soap films
.

Lorentzian surfaces look locally like a two-dimensional slice of relativistic spacetime with one spatial and one time dimension; constant-curvature examples are the flat Lorentzian plane (a two-dimensional subspace of Minkowski space) and the curved de Sitter and anti-de Sitter
planes.

Other types of mathematical planes and surfaces modify or do away with the structures defining the Euclidean plane. For example, the affine plane has a notion of parallel lines but no notion of distance; however, signed areas can be meaningfully compared, as they can in a more general symplectic surface. The projective plane does away with both distance and parallelism. A two-dimensional metric space has some concept of distance but it need not match the Euclidean version. A topological surface can be stretched, twisted, or bent without changing its essential properties. An algebraic surface is a two-dimensional set of solutions of a system of polynomial equations.

Some mathematical spaces have additional arithmetical structure associated with their points. A

hyperbolic number plane, and dual number plane each have points which are considered numbers themselves, and can be added and multiplied. A Riemann surface or Lorentz surface
appear locally like the complex plane or hyperbolic number plane, respectively.

Mathematical spaces are often defined or represented using numbers rather than geometric axioms. One of the most fundamental two-dimensional spaces is the real coordinate space, denoted consisting of pairs of real-number coordinates. Sometimes the space represents arbitrary quantities rather than geometric positions, as in the parameter space of a mathematical model or the configuration space of a physical system.

More generally, other types of numbers can be used as coordinates. The

finite planes, have only a finite set
of elements.

Notes

  1. Riemann surfaces
    .

Further reading
Retrieved from "https://en.wikipedia.org/w/index.php?title=Two-dimensional_space&oldid=1192744260"