For a broader, less mathematical treatment related to this topic, see Space.
"Three-dimensional" redirects here.
For other uses, see 3D (disambiguation).
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).
This is the informal meaning of the term dimension.
When n = 3, 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 ℝ.
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), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane).
In Euclidean geometry
Main article: Coordinate system
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates.
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 are an infinite number of possible methods.
For more, see Euclidean space.
Below are images of the above-mentioned systems.
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 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.
Spheres and balls
Main article: Sphere
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
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.
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.
Main article: Polyhedron
|Class||Platonic solids||Kepler-Poinsot polyhedra|
|Coxeter group||A3, [3,3]||B3, [4,3]||H3, [5,3]|
Surfaces of revolution
Main article: 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.
Main article: Quadric surface
In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
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.
There are six types of non-degenerate quadric surfaces:
- 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 ℝ through that conic that are normal to π).
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.
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
Main article: Dot product
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 ℝ 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:
The magnitude of a vector A is denoted by ||A||.
The dot product of a vector A = [A1, A2, A3] 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 A and B is given by
where θ is the angle between A and B.
Main article: Cross product
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.
Main article: vector calculus
Gradient, divergence and curl
In a rectangular coordinate system, the gradient is given by
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.
This expands as follows:
Line integrals, surface integrals, and volume integrals
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
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.
Fundamental theorem of line integrals
Main article: Fundamental theorem of line integrals
Main article: Stokes' theorem
Main article: Divergence theorem
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.
(dS may be used as a shorthand for ndS.)
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 finite geometry
Many ideas of dimension can be tested with finite geometry.
Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions.
- Dimensional analysis
- Distance from a point to a plane
- Four-dimensional space
- Skew lines § Distance
- Three-dimensional graph
- Two-dimensional space
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Three-dimensional space.