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Ambient space
 
 
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An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object
Physical body

In physics, a physical body is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles ....
.

Mathematics
In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, especially in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, an ambient space is the space surrounding a mathematical object. For example, a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 may be studied in isolation, or it may be studied as an object in two-dimensional space — in which case the ambient space is the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
, or as an object in three-dimensional space
Three-dimensional space

Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth , although any three mutually perpendicular directions can serve as the three dimensions....
 — in which case the ambient space is three-dimensional.






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Encyclopedia


An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object
Physical body

In physics, a physical body is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles ....
.

Mathematics


In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, especially in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, an ambient space is the space surrounding a mathematical object. For example, a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 may be studied in isolation, or it may be studied as an object in two-dimensional space — in which case the ambient space is the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
, or as an object in three-dimensional space
Three-dimensional space

Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth , although any three mutually perpendicular directions can serve as the three dimensions....
 — in which case the ambient space is three-dimensional. To see why this makes a difference, consider the statement "Lines that never meet are necessarily parallel
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
." This is true if the ambient space is two-dimensional, but false if the ambient space is three-dimensional, because in the latter case the lines could be skew lines
Skew lines

In solid geometry, skew lines are two lines that do not intersect but are not parallel. Equivalently, they are lines that are not both in the same plane ....
, rather than parallel.

See also

  • Configuration space
    Configuration space

    Configuration space in physics In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints....
  • Manifold
    Manifold

    In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
     and ambient manifold
  • Submanifold
    Submanifold

    In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties....
    s and Hypersurface
    Hypersurface

    In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
    s
  • Riemannian manifolds
  • Ricci curvature
    Ricci curvature

    In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space....
  • differential form
    Differential form

    In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....


Further reading

  • W H A Schilders, E.J .W. ter Maten, Philippe G. Ciarlet, Numerical Methods in Electromagnetics: Special Volume, Elsevier 2005. (ed., with particular attention to page 120+.)
  • Stephen Wiggins
    Stephen Wiggins

    Stephen Ray Wiggins is an United States applied mathematician, born in Oklahoma City, Oklahoma and best known for his contributions in nonlinear dynamics, chaos theory and nonlinear phenomena, influenced heavily by his PhD advisor Philip Holmes, whom he studied under at Cornell University....
     , Chaotic Transport in Dynamical Systems. 1992. (ed., with particular attention to page 209+.)
  • "Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps" arXiv:0708.3578, 2007
  • "Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps" arXiv:0801.0933, 2008