In Euclidean geometry
Coordinate systemsIn mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axis, coordinate axes are given, each perpendicular to the other two at the Origin (mathematics), 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 numbers, each number giving the distance of that point from the Origin (mathematics), 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 . Below are images of the above-mentioned systems.
Lines and planesTwo distinct points always determine a (straight) Line (geometry), 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 line, parallel or be Skew lines, 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 ℝ3 form a parallelogram, and hence are coplanar.
Spheres and ballsA 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 from a central point . The solid enclosed by the sphere is called a Ball (mathematics), 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.
PolytopesIn three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.
Surfaces of revolutionA Surface (topology), 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 Cone (mathematics), 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 (geometry), cylinder.
Quadric surfacesIn analogy with the conic sections, 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 degeneracy (mathematics), 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 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 (geometry), regulus.
In linear algebraAnother 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 cuboid, 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 coordinate vector, vectors.
Dot product, angle, and lengthA 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 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 .
Cross productThe cross product or vector product is a binary operation on two Euclidean vector, vectors in three-dimensional Euclidean space, 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 (geometry), normal to the plane containing them. It has many applications in mathematics, , and engineering. 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. 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-dimensional cross product, seven dimensions.
Gradient, divergence and curlIn 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 (mathematics), scalar-valued function: : Expanded in Cartesian coordinate system, Cartesian coordinates (see Del in cylindrical and spherical coordinates for Spherical coordinate system, spherical and Cylindrical coordinate system, cylindrical coordinate representations), the curl ∇ × F is, for F composed of [''F''x, ''F''y, ''F''z]: : 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 integralsFor some scalar field ''f'' : ''U'' ⊆ R''n'' → R, the line integral along a piecewise smooth curve ''C'' ⊂ ''U'' is defined as : where r: [a, b] → ''C'' is an arbitrary bijective parametric equation, 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'' → 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: [a, b] → ''C'' is a bijective parametric equation, 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 integrals to integration over Surface (topology), surfaces. 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 Coordinate system, parameterize the surface of interest, ''S'', by considering a system of curvilinear coordinates on ''S'', like the Geographic coordinate system, latitude and longitude on a sphere. Let such a parameterization be x(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the Cartesian coordinate system#Cartesian coordinates in two dimensions, plane. Then, the surface integral is given by : where the expression between bars on the right-hand side is the Magnitude (mathematics), magnitude of the cross product of the partial derivatives of x(''s'', ''t''), and is known as the surface volume element, 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- dimensional domain. It can also mean a multiple integral, triple integral within a region ''D'' in R3 of a function (mathematics), function and is usually written as: :
Fundamental theorem of line integralsThe 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. Let . Then :
Stokes' theoremStokes' theorem relates the surface integral of the Curl (mathematics), 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 theoremSuppose is a subset of (in the case of represents a volume in 3D space) which is compact space, compact and has a piecewise smooth surface, smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the divergence theorem says: : 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 Surface normal, normals, and is the outward pointing unit normal field of the boundary . ( may be used as a shorthand for .)
In topologyThree-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 theory, knot in a piece of string. In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble .
In finite geometryMany 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 (geometry), regulus.Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry'', page 72, Cambridge University Press
See also* Dimensional analysis * Distance from a point to a plane * Four-dimensional space * * Graph of a function of two variables, Three-dimensional graph * Two-dimensional space
References* * George B. Arfken, Arfken, George B. and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). . *
External links* *