Contents 1 In euclidean geometry 1.1 Coordinate systems 1.2 Lines and planes 1.3 Spheres and balls 1.4 Polytopes 1.5 Surfaces of revolution 1.6 Quadric surfaces 2 In linear algebra 2.1 Dot product, angle, and length 2.2 Cross product 3 In calculus 3.1 Gradient, divergence and curl 3.2 Line integrals, surface integrals, and volume integrals 3.3 Fundamental theorem of line integrals 3.4 Stokes' theorem 3.5 Divergence theorem 4 In topology 5 See also 6 Notes 7 References 8 External links In euclidean geometry[edit] Coordinate systems[edit] Main article: Coordinate system 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 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.[1] 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. See Euclidean space. Below are images of the above-mentioned systems. Cartesian coordinate system Cylindrical coordinate system Spherical coordinate system Lines and planes[edit]
Two distinct points always determine a (straight) line. Three distinct
points are either collinear or determine a unique plane. 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
(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.
A two-dimensional perspective projection of a 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 V = 4 3 π r 3 displaystyle V= frac 4 3 pi r^ 3 . 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 ℝ4. If a point has coordinates, P(x, y, z, w), then x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. Polytopes[edit] Main article: Polyhedron In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. Regular polytopes in three dimensions Class Platonic solids Kepler-Poinsot polyhedra Symmetry Td Oh Ih Coxeter group A3, [3,3] B3, [4,3] H3, [5,3] Order 24 48 120 Regular polyhedron 3,3 4,3 3,4 5,3 3,5 5/2,5 5,5/2 5/2,3 3,5/2 Surfaces of revolution[edit] Main article: Surface 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 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, the surface of revolution is a circular cylinder. Quadric surfaces[edit] 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, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , displaystyle Ax^ 2 +By^ 2 +Cz^ 2 +Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0, 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.[2] There are six types of 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 π
and all the lines of ℝ3 through that conic that are normal to
π).[2] 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.[3] Each family is called a regulus.
In linear algebra[edit]
Another way of viewing three-dimensional space is found in linear
algebra, where the idea of independence is crucial.
A ⋅ B = A 1 B 1 + A 2 B 2 + A 3 B 3 . displaystyle mathbf A cdot mathbf B =A_ 1 B_ 1 +A_ 2 B_ 2 +A_ 3 B_ 3 . The magnitude of a vector A is denoted by A. The dot product of a vector A = [A1, A2, A3] with itself is A ⋅ A = ‖ A ‖ 2 = A 1 2 + A 2 2 + A 3 2 , displaystyle mathbf A cdot mathbf A =mathbf A ^ 2 =A_ 1 ^ 2 +A_ 2 ^ 2 +A_ 3 ^ 2 , which gives ‖ A ‖ = A ⋅ A = A 1 2 + A 2 2 + A 3 2 , displaystyle mathbf A = sqrt mathbf A cdot mathbf A = sqrt A_ 1 ^ 2 +A_ 2 ^ 2 +A_ 3 ^ 2 , the formula for the
A ⋅ B = ‖ A ‖ ‖ B ‖ cos θ , displaystyle mathbf A cdot mathbf B =mathbf A ,mathbf B cos theta , where θ is the angle between A and B.
Cross product[edit]
Main article: Cross product
The cross product or vector product is a binary operation on two
vectors 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 nor associative, but is a
The cross-product in respect to a right-handed coordinate system In calculus[edit] Main article: vector calculus Gradient, divergence and curl[edit] In a rectangular coordinate system, the gradient is given by ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k displaystyle nabla f= frac partial f partial x mathbf i + frac partial f partial y mathbf j + frac partial f partial z mathbf k The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function: div F = ∇ ⋅ F = ∂ U ∂ x + ∂ V ∂ y + ∂ W ∂ z . displaystyle operatorname div ,mathbf F =nabla cdot mathbf F = frac partial U partial x + frac partial V partial y + frac partial W partial z . 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]:
i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z
displaystyle begin vmatrix mathbf i &mathbf j &mathbf k \\ frac partial partial x & frac partial partial y & frac partial partial z \\F_ x &F_ y &F_ z end vmatrix where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[7] ( ∂ F z ∂ y − ∂ F y ∂ z ) i + ( ∂ F x ∂ z − ∂ F z ∂ x ) j + ( ∂ F y ∂ x − ∂ F x ∂ y ) k displaystyle left( frac partial F_ z partial y - frac partial F_ y partial z right)mathbf i +left( frac partial F_ x partial z - frac partial F_ z partial x right)mathbf j +left( frac partial F_ y partial x - frac partial F_ x partial y right)mathbf k Line integrals, surface integrals, and volume integrals[edit] For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as ∫ C f d s = ∫ a b f ( r ( t ) )
r ′ ( t )
d t . displaystyle int limits _ C f,ds=int _ a ^ b f(mathbf r (t))mathbf r '(t),dt. where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and a < b displaystyle a<b . For a vector field F : U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as ∫ C F ( r ) ⋅ d r = ∫ a b F ( r ( t ) ) ⋅ r ′ ( t ) d t . displaystyle int limits _ C mathbf F (mathbf r )cdot ,dmathbf r =int _ a ^ b mathbf F (mathbf r (t))cdot mathbf r '(t),dt. 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. A surface integral is a generalization of multiple integrals to integration over 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 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 ∬ S f d S = ∬ T f ( x ( s , t ) ) ‖ ∂ x ∂ s × ∂ x ∂ t ‖ d s d t displaystyle iint _ S f,mathrm d S=iint _ T f(mathbf x (s,t))left partial mathbf x over partial s times partial mathbf x over partial t rightmathrm d s,mathrm d t 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. 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 triple integral within a region D in R3 of a function f ( x , y , z ) , displaystyle f(x,y,z), and is usually written as: ∭ D f ( x , y , z ) d x d y d z . displaystyle iiint limits _ D f(x,y,z),dx,dy,dz. Fundamental theorem of line integrals[edit] Main article: 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. Let φ : U ⊆ R n → R displaystyle varphi :Usubseteq mathbb R ^ n to mathbb R . Then φ ( q ) − φ ( p ) = ∫ γ [ p , q ] ∇ φ ( r ) ⋅ d r . displaystyle varphi left(mathbf q right)-varphi left(mathbf p right)=int _ gamma [mathbf p ,,mathbf q ] nabla varphi (mathbf r )cdot dmathbf r . Stokes' theorem[edit]
Main article: Stokes' theorem
∬ Σ ∇ × F ⋅ d Σ = ∮ ∂ Σ F ⋅ d r . displaystyle iint _ Sigma nabla times mathbf F cdot mathrm d mathbf Sigma =oint _ partial Sigma mathbf F cdot mathrm d mathbf r . Divergence theorem[edit] Main article: Divergence theorem Suppose V is a subset of R n displaystyle mathbb R ^ n (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂V = S ). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:[8] ∭ V ( ∇ ⋅ F ) d V = displaystyle iiint _ V left(mathbf nabla cdot mathbf F right),dV= S displaystyle scriptstyle S ( F ⋅ n ) d S . displaystyle (mathbf F cdot mathbf n ),dS. 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.)
In topology[edit]
R 3 displaystyle scriptstyle mathbb R ^ 3 , the topologists locally model all other 3-manifolds. Wikipedia's globe logo in 3D. Click for 3D manipulation controls See also[edit] Dimensional analysis
Distance from a point to a plane
Notes[edit] ^ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). Calculus : Single and Multivariable (6 ed.). John wiley. ISBN 978-0470-88861-2. ^ a b Brannan, Esplen & Gray 1999, pp. 34–5 ^ Brannan, Esplen & Gray 1999, pp. 41–2 ^ Anton 1994, p. 133 ^ Anton 1994, p. 131 ^ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space. ^ Arfken, p. 43. ^ M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum’s Outlines (2nd ed.). USA: McGraw Hill. ISBN 978-0-07-161545-7. ^ Rolfsen, Dale (1976). Knots and Links. Berkeley, California: Publish or Perish. ISBN 0-914098-16-0. References[edit] Anton, Howard (1994), Elementary Linear Algebra (7th ed.), John Wiley & Sons, ISBN 978-0-471-58742-2 Arfken, George B. and Hans J. Weber. Mathematical Methods For Physicists, Academic Press; 6 edition (June 21, 2005). ISBN 978-0-12-059876-2. Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999), Geometry, Cambridge University Press, ISBN 978-0-521-59787-6 External links[edit] Wikiquote has quotations related to: Three-dimensional space Wikimedia Commons has media related to 3D. The dictionary definition of three-dimensional at Wiktionary Weisstein, Eric W. "Four-Dimensional Geometry". MathWorld. Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry Keith Matthews from University of Queensland, 1991 v t e Dimension Dimensional spaces Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime Other dimensions Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex Dimensions by number Zero One Two Three Four Five Six Seven Eight Nine n-dimensions Negative dimensions |