Three-dimensional space
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (April 2016) |
In
Technically, a
History
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular Platonic solids in a sphere.
In the 17th century, three-dimensional space was described with
In the 19th century, developments of the geometry of three-dimensional space came with
It was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures.
Also during the 19th century came developments in the abstract formalism of vector spaces, with the work of Hermann Grassmann and Giuseppe Peano, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.
In Euclidean geometry
Coordinate systems
plane |
---|
Geometers |
In mathematics,
Other popular methods of describing the location of a point in three-dimensional space include
Below are images of the above-mentioned systems.
Lines and planes
Two distinct points always determine a (straight)
Two distinct lines can either intersect, be
Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.
A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.
Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, and hence are coplanar.
Spheres and balls
A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball).
The volume of the ball is given by
This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.
Polytopes
In three dimensions, there are nine regular polytopes: the five convex
Class | Platonic solids | Kepler-Poinsot polyhedra
| |||||||
---|---|---|---|---|---|---|---|---|---|
Symmetry
|
Td | Oh | Ih | ||||||
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} |
Surfaces of revolution
A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular
Quadric surfaces
In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
There are six types of non-degenerate quadric surfaces:
- Ellipsoid
- Hyperboloid of one sheet
- Hyperboloid of two sheets
- Elliptic cone
- Elliptic paraboloid
- Hyperbolic paraboloid
The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R3 through that conic that are normal to π).[6] Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family.[7] Each family is called a regulus.
In linear algebra
Another 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
A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in ℝ3 can be represented by an ordered triple of real numbers. These numbers are called the components of the vector.
The dot product of two vectors A = [A1, A2, A3] and B = [B1, B2, B3] is defined as:[8]
The magnitude of a vector A is denoted by ||A||. The dot product of a vector A = [A1, A2, A3] with itself is
which gives
the formula for the
Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by[9]
where θ is the angle between A and B.
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.
In function language, the cross product is a function .
The components of the cross product are , and can also be written in components, using Einstein summation convention as where is the Levi-Civita symbol. It has the property that .
Its magnitude is related to the angle between and by the identity
The space and product form an
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.[10]
Abstract description
It can be useful to describe three-dimensional space as a three-dimensional vector space over the real numbers. This differs from in a subtle way. By definition, there exists a basis for . This corresponds to an isomorphism between and : the construction for the isomorphism is found here. However, there is no 'preferred' or 'canonical basis' for .
On the other hand, there is a preferred basis for , which is due to its description as a Cartesian product of copies of , that is, . This allows the definition of canonical projections, , where . For example, . This then allows the definition of the standard basis defined by
Therefore can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, can be obtained by starting with and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.
As opposed to a general vector space , the space is sometimes referred to as a coordinate space.[11]
Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space.
Computationally, it is necessary to work with the more concrete description in order to do concrete computations.
Affine description
A more abstract description still is to model physical space as a three-dimensional affine space over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of , the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces.[12]
This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.
Inner product space
The above discussion does not involve the
In calculus
Gradient, divergence and curl
In a rectangular coordinate system, the gradient of a (differentiable) function is given by
and in index notation is written
The divergence of a (differentiable) vector field F = U i + V j + W k, that is, a function , is equal to the scalar-valued function:
In index notation, with
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 [Fx, Fy, Fz]:
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[13]
In index notation, with Einstein summation convention this is
Line, surface, and volume integrals
For some
where r: [a, b] → C is an arbitrary
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
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 is an integral over a three-dimensional domain or region. When the
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
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
The left side is a
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.[16]
In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble .
In finite geometry
Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one regulus.[17]
See also
- 3D rotation
- Dimensional analysis
- Distance from a point to a plane
- Four-dimensional space
- Skew lines § Distance
- Three-dimensional graph
- Solid geometry
Notes
- ^ a b "IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-19.
- ^ "Euclidean space - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-12.
- ^ "Details for IEV number 113-01-02: "space"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-11-07.
- ^ "Euclidean space | geometry". Encyclopedia Britannica. Retrieved 2020-08-12.
- ISBN 978-0470-88861-2.
- ^ a b Brannan, Esplen & Gray 1999, pp. 34–35
- ^ Brannan, Esplen & Gray 1999, pp. 41–42
- ^ Anton 1994, p. 133
- ^ Anton 1994, p. 131
- 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.
- ^ Lang 1987, ch. I.1
- ^ Berger 1987, Chapter 9.
- ^ Arfken, p. 43.
- ^ "IEC 60050 — Details for IEV number 102-04-40: "volume"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-19.
- ISBN 978-0-07-161545-7.
- ISBN 0-914098-16-0.
- ISBN 0-521-48277-1
References
- Anton, Howard (1994), Elementary Linear Algebra (7th ed.), John Wiley & Sons, ISBN 978-0-471-58742-2
- ISBN 978-0-12-059876-2.
- ISBN 3-540-11658-3
- Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999), Geometry, Cambridge University Press, ISBN 978-0-521-59787-6
- Lang, Serge (1987), Linear algebra (3rd ed.), Springer, ISBN 978-1-4757-1949-9
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