Definitions

1. Three-dimensional space is a geometric model of the physical universe in which we live.
2. In three-dimensional space, a hyperplane is an ordinary plane; it divides the space into two half-spaces. (Web site)
3. A basis in three-dimensional space is a set of three linearly independent vectors, called basis vectors.

Four-Dimensional Space

1. In four-dimensional space there exist exactly six regular polytopes, five of them generalizations from three-dimensional space. (Web site)

Form Vector Spaces

1. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

Position

1. A change in the position of a particle in three-dimensional space can be completely specified by three coordinates.

Versions

1. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. (Web site)

Topological Space

1. A topological space is a generalization of the notion of an object in three-dimensional space.

Squares

1. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the three-dimensional space.
2. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

Boundaries

1. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. (Web site)

Atoms

1. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. (Web site)
2. These two versions of the molecule are not occupying the same three-dimensional space, even though they contain the same basic set of atoms. (Web site)

Points

1. A two-dimensional geometric figure (a collection of points) in three-dimensional space. (Web site)

Parallel

1. In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space.

Polyhedron

1. The polyhedron surrounds a volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron. (Web site)

Line

1. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space in which we live. (Web site)
2. A parallel combination of a line and a plane may be located in the same three-dimensional space. (Web site)

Mathematically

1. Such lipid bilayers are mathematically described as two-dimensional surfaces embedded in a three-dimensional space.

Embedding

1. Since these are often realizable as an embedding of three-dimensional space, it is possible to literally picture what is going on mathematically. (Web site)

Sphere

1. Analogously: the n -dimensional sphere Sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. (Web site)

Product

1. The cross product is a marvelous tool to use for such problems as this in three-dimensional space.

Smooth Surfaces

1. Current technology can cope well with curves in the plane and smooth surfaces in three-dimensional space. (Web site)

Unit Circle

1. The simplest form of knot theory involves the embedding of the unit circle into three-dimensional space. (Web site)

Point

1. The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point. (Web site)

Vector Spaces

1. Vector Spaces generalises the plane and three-dimensional space, providing a common structure for studying seemingly different problems. (Web site)

Rotation Group

1. There is a close analogy between this group and O(3), the rotation group in three-dimensional space. (Web site)

Projective Geometry

1. The concepts of a point and a plane in three-dimensional space are dual in projective geometry; the concept of a straight line is dual to itself.

Mathematics

1. In mathematics, cardinal directions or cardinal points are the six principal directions or points along the x-, y- and z- axis of Three-dimensional space.

Polynomial Functions

1. Abstract The paper concerns analytical integration of polynomial functions over linear polyhedra in three-dimensional space. (Web site)
2. Symbolic integration of polynomial functions over a linear polyhedron in Euclidean three-dimensional space. (Web site)

Level Surface

1. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. (Web site)

Spherical Coordinates

1. Here Ip, 2ψ and 2χ are the spherical coordinates of the polarization state in the three-dimensional space of the last three Stokes parameters.
2. Spherical Coordinates: A coordinate system in which points in three-dimensional space are given by two angles and a distance from the origin.

Set

1. Using this mode of thought, a line in three-dimensional space is the same as the set of points on the line, and is therefore just a subset of R 3.
2. This implies, that the polyhedron can be defined as a set of polygons limiting a part of three-dimensional space.

Surfaces

1. Surfaces are tangible in three-dimensional space only as the boundaries of three-dimensional solid objects. (Web site)
2. Manifolds include familiar curves such as the circle, or surfaces in three-dimensional space that are locally smooth.

Surface

1. Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. (Web site)

Mechanics

1. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
2. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. (Web site)
3. Many problems, particularly in mechanics, involve the use of two- or three-dimensional space to describe where objects are and what they are doing. (Web site)

Two-Dimensional

1. These systems can represent points in two-dimensional or three-dimensional space. (Web site)

Rotation

1. The 3×3 rotation matrix corresponds to a rotation of approximately 74° around the axis (− 1 ⁄ 3, 2 ⁄ 3, 2 ⁄ 3) in three-dimensional space.

Rotations

1. The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space.
2. This study addresses the way the three-dimensional space of rotations is also embedded in the group structure of neural connectivity. (Web site)
3. Rotations in ordinary three-dimensional space differ than those in two dimensions in a number of important ways. (Web site)

Two-Dimensional Space

1. It springs from the idea that any point in two-dimensional space can be represented by two numbers and any point in three-dimensional space by three. (Web site)

Space

1. Those three numbers are usually thought of as being in a "color space" which is a kind of abstract space like three-dimensional space.

Embedded

1. Like the surface of a ball embedded in three-dimensional space, we can imagine four-dimensional spacetime as embedded in a flat space of a higher dimension. (Web site)
2. Like the surface of a ball embedded in three-dimensional space, we can imagine four dimensional spacetime as embedded in a flat space of a higher dimension.
3. The projective plane cannot strictly be embedded (that is without intersection) in three-dimensional space.

Vectors

1. Suppose we have a rotation R, defined in three-dimensional space, that rotates vectors by angle theta about unit vector n. (Web site)

Unit Sphere

1. The unit sphere in three-dimensional space R 3 is the set of points (x, y, z) such that x 2 + y 2 + z 2 = 1. (Web site)

Planes

1. For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes.
2. These two equations correspond to two planes in three-dimensional space that intersect in some line which passes through the origin of the coordinate system.

Plane

1. For example, the function which maps the point (x, y, z) in three-dimensional space R 3 to the point (x, y, 0) is a projection onto the x - y plane.
2. In a three-dimensional space, another important way of defining a plane is by specifying a point and a normal vector to the plane. (Web site)
3. In other words, a hyperplane is a higher-dimensional analog of a (two-dimensional) plane in three-dimensional space. (Web site)

Orientable

1. The whole two-dimensional x - y plane, thought of as a subset of three-dimensional space, is also orientable: we can distinguish between "above" and "below". (Web site)

Klein Bottle

1. The corresponding trivial bundle would be a torus, S 1 × S 1. The Klein bottle immersed in three-dimensional space. (Web site)
2. Klein bottle The Klein bottle immersed in three-dimensional space. (Web site)

Three-Dimensional Space

1. Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space.
2. In three-dimensional space, it has two distinct forms, which are mirror images (or " enantiomorphs ") of each other.
3. In mathematics, the cross product, vector product or Gibbs vector product is a binary operation on two vectors in three-dimensional space.

Categories

1. Klein Bottle
2. Plane
3. Unit Sphere
4. Mirror Images
5. Embedded
6. Books about "Three-Dimensional Space" in Amazon.com
   

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