Three-dimensional space

For a broader, less mathematical treatment related to this topic, see Space. Three-dimensional space_sentence_0

"Three-dimensional" redirects here. Three-dimensional space_sentence_1

For other uses, see 3D (disambiguation). Three-dimensional space_sentence_2

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). Three-dimensional space_sentence_3

This is the informal meaning of the term dimension. Three-dimensional space_sentence_4

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. Three-dimensional space_sentence_5

When n = 3, the set of all such locations is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). Three-dimensional space_sentence_6

It is commonly represented by the symbol ℝ. Three-dimensional space_sentence_7

This serves as a three-parameter model of the physical universe (that is, the spatial part, without considering time), in which all known matter exists. Three-dimensional space_sentence_8

While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifolds. Three-dimensional space_sentence_9

In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Three-dimensional space_sentence_10

Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and length. Three-dimensional space_sentence_11

In Euclidean geometry Three-dimensional space_section_0

Coordinate systems Three-dimensional space_section_1

Main article: Coordinate system Three-dimensional space_sentence_12

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three-dimensional space_sentence_13

Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. Three-dimensional space_sentence_14

They are usually labeled x, y, and z. Three-dimensional space_sentence_15

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. Three-dimensional space_sentence_16

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. Three-dimensional space_sentence_17

For more, see Euclidean space. Three-dimensional space_sentence_18

Below are images of the above-mentioned systems. Three-dimensional space_sentence_19

Three-dimensional space_unordered_list_0

• Three-dimensional space_item_0_0
• Three-dimensional space_item_0_1
• Three-dimensional space_item_0_2

Lines and planes Three-dimensional space_section_2

Two distinct points always determine a (straight) line. Three-dimensional space_sentence_20

Three distinct points are either collinear or determine a unique plane. Three-dimensional space_sentence_21

On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space. Three-dimensional space_sentence_22

Two distinct lines can either intersect, be parallel or be skew. Three-dimensional space_sentence_23

Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Three-dimensional space_sentence_24

Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three-dimensional space_sentence_25

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. Three-dimensional space_sentence_26

In the last case, the three lines of intersection of each pair of planes are mutually parallel. Three-dimensional space_sentence_27

A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. Three-dimensional space_sentence_28

In the last case, there will be lines in the plane that are parallel to the given line. Three-dimensional space_sentence_29

A hyperplane is a subspace of one dimension less than the dimension of the full space. Three-dimensional space_sentence_30

The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. Three-dimensional space_sentence_31

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. Three-dimensional space_sentence_32

A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Three-dimensional space_sentence_33

Varignon's theorem states that the midpoints of any quadrilateral in ℝ form a parallelogram, and hence are coplanar. Three-dimensional space_sentence_34

Spheres and balls Three-dimensional space_section_3

Main article: Sphere Three-dimensional space_sentence_35

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). Three-dimensional space_sentence_36

The volume of the ball is given by Three-dimensional space_sentence_37

Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ. Three-dimensional space_sentence_38

If a point has coordinates, P(x, y, z, w), then x + y + z + w = 1 characterizes those points on the unit 3-sphere centered at the origin. Three-dimensional space_sentence_39

Polytopes Three-dimensional space_section_4

Main article: Polyhedron Three-dimensional space_sentence_40

In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. Three-dimensional space_sentence_41

Three-dimensional space_table_general_0

Regular polytopes in three dimensionsThree-dimensional space_table_caption_0
OrderThree-dimensional space_header_cell_0_3_0 24Three-dimensional space_cell_0_3_1 48Three-dimensional space_cell_0_3_2 120Three-dimensional space_cell_0_3_4
Regular polyhedronThree-dimensional space_header_cell_0_4_0 {3,3}Three-dimensional space_cell_0_4_1 {4,3}Three-dimensional space_cell_0_4_2 {3,4}Three-dimensional space_cell_0_4_3 {5,3}Three-dimensional space_cell_0_4_4 {3,5}Three-dimensional space_cell_0_4_5 {5/2,5}Three-dimensional space_cell_0_4_6 {5,5/2}Three-dimensional space_cell_0_4_7 {5/2,3}Three-dimensional space_cell_0_4_8 {3,5/2}Three-dimensional space_cell_0_4_9

Surfaces of revolution Three-dimensional space_section_5

Main article: Surface of revolution Three-dimensional space_sentence_42

A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. Three-dimensional space_sentence_43

The plane curve is called the generatrix of the surface. Three-dimensional space_sentence_44

A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Three-dimensional space_sentence_45

Simple examples occur when the generatrix is a line. Three-dimensional space_sentence_46

If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection. Three-dimensional space_sentence_47

However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder. Three-dimensional space_sentence_48

Main article: Quadric surface Three-dimensional space_sentence_49

In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, Three-dimensional space_sentence_50

where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero, is called a quadric surface. Three-dimensional space_sentence_51

There are six types of non-degenerate quadric surfaces: Three-dimensional space_sentence_52

Three-dimensional space_ordered_list_1

1. EllipsoidThree-dimensional space_item_1_3
2. Hyperboloid of one sheetThree-dimensional space_item_1_4
3. Hyperboloid of two sheetsThree-dimensional space_item_1_5
4. Elliptic coneThree-dimensional space_item_1_6
5. Elliptic paraboloidThree-dimensional space_item_1_7
6. Hyperbolic paraboloidThree-dimensional space_item_1_8

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 ℝ through that conic that are normal to π). Three-dimensional space_sentence_53

Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Three-dimensional space_sentence_54

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. Three-dimensional space_sentence_55

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. Three-dimensional space_sentence_56

Each family is called a regulus. Three-dimensional space_sentence_57

In linear algebra Three-dimensional space_section_7

Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Three-dimensional space_sentence_58

Space has three dimensions because the length of a box is independent of its width or breadth. Three-dimensional space_sentence_59

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. Three-dimensional space_sentence_60

Dot product, angle, and length Three-dimensional space_section_8

Main article: Dot product Three-dimensional space_sentence_61

A vector can be pictured as an arrow. Three-dimensional space_sentence_62

The vector's magnitude is its length, and its direction is the direction the arrow points. Three-dimensional space_sentence_63

A vector in ℝ can be represented by an ordered triple of real numbers. Three-dimensional space_sentence_64

These numbers are called the components of the vector. Three-dimensional space_sentence_65

The dot product of two vectors A = [A1, A2, A3] and B = [B1, B2, B3] is defined as: Three-dimensional space_sentence_66

The magnitude of a vector A is denoted by ||A||. Three-dimensional space_sentence_67

The dot product of a vector A = [A1, A2, A3] with itself is Three-dimensional space_sentence_68

which gives Three-dimensional space_sentence_69

the formula for the Euclidean length of the vector. Three-dimensional space_sentence_70

Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by Three-dimensional space_sentence_71

where θ is the angle between A and B. Three-dimensional space_sentence_72

Cross product Three-dimensional space_section_9

Main article: Cross product Three-dimensional space_sentence_73

The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. Three-dimensional space_sentence_74

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. Three-dimensional space_sentence_75

It has many applications in mathematics, physics, and engineering. Three-dimensional space_sentence_76

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. Three-dimensional space_sentence_77

One can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. Three-dimensional space_sentence_78

But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. Three-dimensional space_sentence_79

In calculus Three-dimensional space_section_10

Main article: vector calculus Three-dimensional space_sentence_80

Gradient, divergence and curl Three-dimensional space_section_11

In a rectangular coordinate system, the gradient is given by Three-dimensional space_sentence_81

The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function: Three-dimensional space_sentence_82

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]: Three-dimensional space_sentence_83

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. Three-dimensional space_sentence_84

This expands as follows: Three-dimensional space_sentence_85

Line integrals, surface integrals, and volume integrals Three-dimensional space_section_12

For some scalar field f : U ⊆ R → R, the line integral along a piecewise smooth curve C ⊂ U is defined as Three-dimensional space_sentence_86

For a vector field F : U ⊆ R → R, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as Three-dimensional space_sentence_87

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. Three-dimensional space_sentence_88

A surface integral is a generalization of multiple integrals to integration over surfaces. Three-dimensional space_sentence_89

It can be thought of as the double integral analog of the line integral. Three-dimensional space_sentence_90

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. Three-dimensional space_sentence_91

Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Three-dimensional space_sentence_92

Then, the surface integral is given by Three-dimensional space_sentence_93

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. Three-dimensional space_sentence_94

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. Three-dimensional space_sentence_95

A volume integral refers to an integral over a 3-dimensional domain. Three-dimensional space_sentence_96

Fundamental theorem of line integrals Three-dimensional space_section_13

Main article: Fundamental theorem of line integrals Three-dimensional space_sentence_97

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. Three-dimensional space_sentence_98

Stokes' theorem Three-dimensional space_section_14

Main article: Stokes' theorem Three-dimensional space_sentence_99

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: Three-dimensional space_sentence_100

Divergence theorem Three-dimensional space_section_15

Main article: Divergence theorem Three-dimensional space_sentence_101

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 ∂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 ∂V. Three-dimensional space_sentence_102

(dS may be used as a shorthand for ndS.) Three-dimensional space_sentence_103

In topology Three-dimensional space_section_16

Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. Three-dimensional space_sentence_104

For example, at least three dimensions are required to tie a knot in a piece of string. Three-dimensional space_sentence_105

In finite geometry Three-dimensional space_section_17

Many ideas of dimension can be tested with finite geometry. Three-dimensional space_sentence_106

The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. Three-dimensional space_sentence_107

It is an instance of Galois geometry, a study of projective geometry using finite fields. Three-dimensional space_sentence_108

Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. Three-dimensional space_sentence_109

For example, any three skew lines in PG(3,q) are contained in exactly one regulus. Three-dimensional space_sentence_110