Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s'') are required to determine the
position of an element (i.e.,
point). This is the informal meaning of the term
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
.
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of
numbers can be understood as the
Cartesian coordinates of a location in a -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. The set of these -tuples is commonly denoted
and can be identified to the -dimensional Euclidean space.
When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
(when
relativity theory
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena ...
is not considered), in which all known
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
exists. 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. In this classical example, when the three values refer to measurements in different directions (
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
), any three directions can be chosen, provided that
vectors in these directions do not all lie in the same
2-space (
plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms ''
width
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
/breadth'', ''
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
/depth'', and ''
length''.
History
Books XI to XIII of
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
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
Cartesian coordinates, with the advent of
analytic geometry developed by
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
in his work ''
La Géométrie'' and
Pierre de Fermat in the manuscript ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.
In the 19th century, developments of the geometry of three-dimensional space came with
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
's development of the
quaternions. In fact, it was Hamilton who coined the terms
scalar and
vector, and they were first defined within
his geometric framework for quaternions. Three dimensional space could then be described by quaternions
which had vanishing scalar component, that is,
. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements
, as well as the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
and
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.
It wasn't until
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
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
In mathematics,
analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three
coordinate axes
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
are given, each perpendicular to the other two at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, the point at which they cross. They are usually labeled , and . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, each number giving the distance of that point from the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
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 are an infinite number of possible methods. For more, see
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.
Below are images of the above-mentioned systems.
Image:Coord XYZ.svg, Cartesian coordinate system
Image:Cylindrical Coordinates.svg, Cylindrical coordinate system
Image:Spherical Coordinates (Colatitude, Longitude).svg, Spherical coordinate system
Lines and planes
Two distinct points always determine a (straight)
line. Three distinct points are either
collinear or determine a unique plane. On the other hand, four distinct points can either be collinear,
coplanar, or determine the entire space.
Two distinct lines can either intersect, be
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
or be
skew. 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.
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 ℝ
3 form a
parallelogram, and hence are coplanar.
Spheres and balls
A
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
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 from a central point . The solid enclosed by the sphere is called a
ball (or, more precisely a 3-ball).
The volume of the ball is given by
and the surface area of the sphere is
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 . If a point has coordinates, , then characterizes those points on the unit 3-sphere centered at the origin.
This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3D space.
Polytopes
In three dimensions, there are nine regular polytopes: the five convex
Platonic solids and the four nonconvex
Kepler-Poinsot polyhedra.
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
In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent.
Examples
A cone can ...
'' 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
cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an ...
.
Quadric surfaces
In analogy with the
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
:
where and are real numbers and not all of and are zero, is called a quadric surface.
There are six types of
non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...
quadric surfaces:
#
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
#
Hyperboloid of one sheet
#
Hyperboloid of two sheets
#
Elliptic cone
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the '' ...
#
Elliptic paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every pl ...
#
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 and all the lines of through that conic that are normal to ).
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. Each family is called a
regulus
Regulus is the brightest object in the constellation Leo and one of the brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Re ...
.
In linear algebra
Another way of viewing three-dimensional space is found in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, 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 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 and is defined as:
:
The magnitude of a vector is denoted by . The dot product of a vector with itself is
:
which gives
:
the formula for the
Euclidean length
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
of the vector.
Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors and is given by
:
where is the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
between and .
Cross product
The
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
or vector product is a
binary operation on two
vectors in three-dimensional
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to both and therefore
normal to the plane containing them. It has many applications in mathematics,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, and
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
.
In function language, the cross product is a function
.
The components of the cross product are