三次元の

関: 3D、third dimension、three dimension、tridimensional

WordNet

involving or relating to three dimensions or aspects; giving the illusion of depth; "lifelike three-dimensional characters"; "a three-dimensional account of conditions under the new government"; "they shot the movie in three-D" (同)3-dimensional, third-dimensional, three-d
the property of having three dimensions (同)third-dimensionality
indicate the dimensions on; "These techniques permit us to dimension the human heart"
shape or form to required dimensions
the magnitude of something in a particular direction (especially length or width or height)
one of three Cartesian coordinates that determine a position in space
being one more than two (同)3, iii
the cardinal number that is the sum of one and one and one (同)3, III, trio, threesome, tierce, leash, troika, triad, trine, trinity, ternary, ternion, triplet, tercet, terzetto, trey, deuce-ace
of or relating to dimensions
having dimension--the quality or character or stature proper to a person; "never matures as a dimensional character; he is pasty, bland, faceless"- Norman Cousins

PrepTutorEJDIC

三次元の,立体の;立体感のある
〈C〉(長さ・幅・厚さなどの)『寸法』 / 《通例複数形で》大きさ,範囲,規模 / 〈C〉広がり,展望(scope) / 〈C〉(数学で)次元
『3』;3の記号(3,?など) / 〈U〉3時,3分;3歳 / 《複数扱い》3人;3個 / 〈C〉三つ(3人,3個)一組のもの / (カードゲーム・さいころなどの)3の札,3の目 / 『3の』,3人の,3個の / 《補語にのみ用いて》3歳の(で)
(長さ・幅・厚さなどの)寸法の / (数学で)…次元の

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/09/09 01:06:01」(JST)

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[Wiki en表示]

For a broader, less mathematical treatment related to this topic, see Space.

Three-dimensional Cartesian coordinate system with the x-axis pointing towards the observer.

(See diagram description for correction.)

Three-dimensional space (also: tri-dimensional space) is a geometric three-parameter model of the physical universe (without considering time) in which all known matter exists. These three dimensions can be labeled by a combination of three chosen from the terms length, width, height, depth, and breadth. Any three directions can be chosen, provided that they do not all lie in the same plane.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol . This space is only one example of a great variety of spaces in three dimensions called 3-manifolds.

1 In geometry
- 1.1 Coordinate systems
- 1.2 Polytopes
- 1.3 Sphere
- 1.4 Orthogonality
2 In linear algebra
- 2.1 Dot product, angle, and length
- 2.2 Cross product
3 In calculus
- 3.1 Gradient, divergence and curl
- 3.2 Line integrals, surface integrals, and volume integrals
- 3.3 Fundamental theorem of line integrals
- 3.4 Stokes' theorem
- 3.5 Divergence theorem
4 In topology
5 See also
6 References
7 External links

In geometry

Coordinate systems

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there is an infinite number of possible methods. See Euclidean space.

Below are images of the above-mentioned systems.

Cartesian coordinate system
Cylindrical coordinate system
Spherical coordinate system

Polytopes

In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.

Regular polytopes in three dimensions
Class	Platonic solids					Kepler-Poinsot polyhedra
Symmetry	T_d	O_h		I_h
Coxeter group	A₃, [3,3]	B₃, [4,3]		H₃, [5,3]
Order	24	48		120
Regular polyhedron	{3,3}	{4,3}	{3,4}	{5,3}	{3,5}	{5/2,5}	{5,5/2}	{5/2,3}	{3,5/2}

Sphere

A two-dimensional perspective projection of a sphere

A sphere in 3-space (also called a 2-sphere because its surface is 2-dimensional) consists of the set of all points in 3-space at a fixed distance r from a central point P. The volume enclosed by this surface is:

Another type of sphere, but having a three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space at distance one. If any position is , then characterize a point in the 3-sphere.

Orthogonality

In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: north/south (latitude), east/west (longitude) and up/down (altitude). These pairs of directions are mutually orthogonal: They are at right angles to each other. Movement along one axis does not change the coordinate value of the other two axes. In mathematical terms, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.

In linear algebra

Another mathematical way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors.

Dot product, angle, and length

The dot product of two vectors A = [A₁, A₂,A₃] and B = [B₁, B₂,B₃] is defined as:^[1]

A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by^[2]

where θ is the angle between A and B.

The dot product of a vector A by itself is

which gives

the formula for the Euclidean length of the vector.

Cross product

The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product a × b of the vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.

The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

One can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.^[3]

The cross-product in respect to a right-handed coordinate system

In calculus

Gradient, divergence and curl

In a rectangular coordinate system, the gradient is given by

\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}

The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:

\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial U}{\partial x} +\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z }.

Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [F_x, F_y, F_z]:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ F_x & F_y & F_z \end{vmatrix}

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:^[4]

Line integrals, surface integrals, and volume integrals

For some scalar field f : U ⊆ Rⁿ → R, the line integral along a piecewise smooth curve C ⊂ U is defined as

where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and .

For a vector field F : U ⊆ Rⁿ → Rⁿ, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as

where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.

A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by

\iint_{S} f \,\mathrm dS = \iint_{T} f(\mathbf{x}(s, t)) \left\|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right\| \mathrm ds\, \mathrm dt

where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. Given a vector field v on S, that is a function that assigns to each x in S a vector v(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.

A volume integral refers to an integral over a 3-dimensional domain.

It can also mean a triple integral within a region D in R³ of a function and is usually written as:

Fundamental theorem of line integrals

The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let . Then

Stokes' theorem

Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:

Divergence theorem

Suppose $V$ is a subset of (in the case of $n = 3, V$ represents a volume in 3D space) which is compact and has a piecewise smooth boundary $S$ (also indicated with $\partial V = S$ ). If $F$ is a continuously differentiable vector field defined on a neighborhood of $V$ , then the divergence theorem says:^[5]

The left side is a volume integral over the volume $V$ , the right side is the surface integral over the boundary of the volume $V$ . The closed manifold $\partial V$ is quite generally the boundary of $V$ oriented by outward-pointing normals, and $n$ is the outward pointing unit normal field of the boundary $\partial V$ . ( $d S$ may be used as a shorthand for $n dS$ .)

In topology

Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a knot in a piece of string.^[6]

With the space , the topologists locally model all other 3-manifolds.

References

^ S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.
^ M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7.
^ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
^ Arfken, p. 43.
^ M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum’s Outlines (2nd ed.). USA: McGraw Hill. ISBN 978-0-07-161545-7.
^ Rolfsen, Dale (1976). Knots and Links. Berkeley, California: Publish or Perish. ISBN 0-914098-16-0.

External links

The dictionary definition of three-dimensional at Wiktionary
Weisstein, Eric W., "Four-Dimensional Geometry", MathWorld.
Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry Keith Matthews from University of Queensland, 1991

UpToDate Contents

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English Journal

3D reconstruction of biological structures: automated procedures for alignment and reconstruction of multiple tilt series in electron tomography.

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Advanced structural and chemical imaging.Adv Struct Chem Imaging.2017;2:8. Epub 2016 Jun 28.
Transmission electron microscopy allows the collection of multiple views of specimens and their computerized three-dimensional reconstruction and analysis with electron tomography. Here we describe development of methods for automated multi-tilt data acquisition, tilt-series processing, and alignmen
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Electrochemical treatment of water containing Microcystis aeruginosa in a fixed bed reactor with three-dimensional conductive diamond anodes.

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Xu J1, Liu XF2, Wang YT3.
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XPO1 Inhibition Preferentially Disrupts the 3D Nuclear Organization of Telomeres in Tumor Cells.

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Japanese Journal

Clinical Utility of Three-dimensional Computed Tomography for Esophageal Reconstruction using Colon Interposition after Gastrectomy

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The Kitakanto medical journal 62(2), 143-146, 2012-05-01
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The classical origin of quantum affine algebra in squashed sigma models

Kawaguchi Io,Matsumoto Takuya,Yoshida Kentaroh
Journal of High Energy Physics 2012(4), 2012-04
… We consider a quantum affine algebra realized in two-dimensional non-linear sigma models with target space three-dimensional squashed sphere. …
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An automated three-dimensional internal structure observation system based on high-speed serial sectioning of steel materials

Fujisaki Kazuhiro,Yamashita Norio,Yokota Hideo
Precision Engineering 36(2), 315-321, 2012-04
… For three-dimensional observation of the internal structure of hard materials, we developed an automated system based on serial sectioning with precision cutting and optical microscopy. … The three-dimensional shapes and positions of continuously distributed inclusions found deep within the bearing steels, as well as cracks originating from these inclusions, could be detected with resolutions as high as 80 x 80 nm and in a wide field of view using this system. …
NAID 120003994045

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リンク元	「三次元」「third dimension」「3D」「tridimensional」「three dimension」
拡張検索	「three-dimensional imaging」「three-dimensional echocardiography」「three-dimensional image」
関連記事	「dimension」「dimensional」