Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
s) are required to determine the position of an element (i.e., point
). This is the informal meaning of the term dimension
, a sequence
can be understood as a location in -dimensional space. When , the set of all such locations is called three-dimensional Euclidean space
(or simply Euclidean space when the context is clear). It is commonly represented by the symbol . 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. 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-manifold
s. 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
). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms ''width
'', ''depth'', and ''length
In Euclidean geometry
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 , and . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real number
s, 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 are an infinite number of possible methods. For more, see Euclidean space
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
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.
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.
states that the midpoints of any quadrilateral in ℝ3
form a parallelogram
, and hence are coplanar.
Spheres and balls
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
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.
In three dimensions, there are nine regular polytopes: the five convex Platonic solid
s and the four nonconvex Kepler-Poinsot polyhedra
Surfaces of revolution
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 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
In analogy with the conic section
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
# 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 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 surface
s, 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
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 vector
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
the formula for the Euclidean length
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
between and .
The cross product
or vector product is a binary operation
on two vector
s 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
, but is a Lie algebra
with the cross product being the Lie bracket.
One can in ''n'' dimensions take the product of 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
Gradient, divergence and curl
In a rectangular coordinate system, the gradient is given by
The divergence of a continuously differentiable vector field
F = ''U'' i + ''V'' j + ''W'' k is equal to the scalar
Expanded in Cartesian coordinates
(see Del in cylindrical and spherical coordinates
coordinate representations), the curl ∇ × F is, for F composed of 'F''x, ''F''y, ''F''z
where i, j, and k are the unit vector
s for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows:
Line integrals, surface integrals, and volume integrals
For some scalar field
''f'' : ''U'' ⊆ R''n''
→ R, the line integral along a piecewise smooth curve
''C'' ⊂ ''U'' is defined as
where r: , 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'' ⊆ R''n''
, 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: , 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 integral
s to integration over surface
s. 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
where the expression between bars on the right-hand side is the magnitude
of the cross product
of the partial derivative
s 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-dimension
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.
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:
Suppose is a subset of
(in the case of represents a volume in 3D space) which is compact
and has a piecewise smooth boundary
(also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the divergence theorem
The left side is a volume integral
over the volume , the right side is the surface integral
over the boundary of the volume . The closed manifold is quite generally the boundary of oriented by outward-pointing normals
, and is the outward pointing unit normal field of the boundary . ( may be used as a shorthand for .)
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.
In differential geometry
the generic three-dimensional spaces are 3-manifold
s, 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 plane
s as its 2-dimensional subspaces. It is an instance of Galois geometry
, a study of projective geometry
using finite field
s. 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
[Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry'', page 72, Cambridge University Press ]
* Dimensional analysis
* Distance from a point to a plane
* Four-dimensional space
* Three-dimensional graph
* Two-dimensional space
* Arfken, George B.
and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). .
Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry
Keith Matthews from University of Queensland