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ompass constructions angle polytope centroid diagonal orthogonality perpendicular parallel vertex congruence similarity symmetry zero dimensional point one dimensional line line segment ray curve geodesic length two dimensional surface plane area polygon simple convex concave star regular reuleaux triangle centers altitude hypotenuse pythagorean theorem circular reuleaux hyperbolic ideal spherical quadrilateral parallelogram rectangle square rhombus rhomboid trapezoid kite circle radius diameter circumference disk area three dimensional surface area volume polyhedron platonic solid tetrahedron reuleaux cuboid cube octahedron dodecahedron icosahedron pyramid star toroidal ideal solid of revolution sphere great circle cylinder cone four other dimensional 4 polytope simplex 5 cell hypercube tesseract n sphere geometers by name aida aryabhata ahmes alhazen apollonius archimedes atiyah baudhayana bolyai brahmagupta cartan chern coxeter descartes euclid euler gauss gromov hilbert huygens jyeṣṭhadeva kātyāyana khayyám klein lobachevsky manava minkowski minggatu pascal pythagoras parameshvara poincaré riemann sakabe sijzi al tusi veblen virasena yang hui al yasamin zhang list of geometers by period bce ahmes baudhayana manava pythagoras euclid archimedes apollonius 1 1400s zhang kātyāyana aryabhata brahmagupta virasena alhazen sijzi khayyám al yasamin al tusi yang hui parameshvara 1400s 1700s jyeṣṭhadeva descartes pascal huygens minggatu euler sakabe aida 1700s 1900s gauss lobachevsky bolyai riemann klein poincaré hilbert minkowski cartan veblen coxeter chern present day atiyah gromov v t e 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 21 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 22 23 for more 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 on the other hand four distinct points can either be collinear coplanar or determine the entire space 24 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 25 relations between up to three planes only in example 12 do three planes meet to form a point two distinct planes can either meet in a common line or are parallel i e do not meet 25 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 26 a line can lie in a given plane intersect that plane in a unique point or be parallel to the plane 25 in the last case lines can be formed 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 27 varignon s theorem states that the midpoints of any quadrilateral in r 3 displaystyle mathbb r 3 form a parallelogram and hence are coplanar 28 spheres and balls edit main article sphere a perspective projection of a sphere onto two dimensions a sphere in 3 space also called a 2 sphere because like all surfaces it is intrinsically two dimensional 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 3 ball 29 the volume of the ball is given by 30 v 4 3 π r 3 displaystyle v frac 4 3 pi r 3 and the surface area of the sphere is 30 a 4 π r 2 displaystyle a 4 pi r 2 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 4 if a point has coordinates p x y z w then x 2 y 2 z 2 w 2 1 characterizes those points on the unit 3 sphere centered at the origin 31 this 3 sphere is an example of a 3 manifold a space which is looks locally like 3 d space 32 in precise topological terms each point of the 3 sphere has a neighborhood which is homeomorphic to an open subset of 3 d space 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 33 regular polytopes in three dimensions class platonic solids kepler poinsot polyhedra symmetry t d o h i h coxeter group a 3 3 3 b 3 4 3 h 3 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 34 35 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 34 35 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 36 there are six types of non degenerate quadric surfaces 36 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 r 3 through that conic that are normal to π 36 elliptic cones are sometimes considered to be degenerate quadric surfaces as well citation needed 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 of one family intersects with just one exception every member of the other family 36 each family is called a regulus 37 in linear algebra edit in linear algebra the perspective of three dimensional space is crucially dependent on the concept of independence 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 38 dot product angle and length edit 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 3 displaystyle mathbb r 3 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 a 1 a 2 a 3 and b b 1 b 2 b 3 is defined as 39 a b a 1 b 1 a 2 b 2 a 3 b 3 i 1 3 a i b i displaystyle mathbf a cdot mathbf b a_ 1 b_ 1 a_ 2 b_ 2 a_ 3 b_ 3 sum _ i 1 3 a_ i b_ i the magnitude of a vector a is denoted by a the dot product of a vector a a 1 a 2 a 3 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 39 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 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 39 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 for a physical example consider a block on an inclined plane that is being pulled downward by a gravitational force the dot product can be used to compute the work w displaystyle w performed by the constant force vector g displaystyle mathbf g that is applied at an angle θ displaystyle theta to the downslope direction of motion d displaystyle mathbf d that is 40 w g d g d cos θ displaystyle w mathbf g cdot mathbf d mathbf g mathbf d cos theta 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 41 for example it can be used to compute the amount of torque on a bolt being turned by a wrench or the lorentz force on an electron travelling through a magnetic field 42 in function language the cross product is a function r 3 r 3 r 3 displaystyle times mathbb r 3 times mathbb r 3 rightarrow mathbb r 3 43 the cross product in respect to a right handed coordinate system the components of the cross product are a b a 2 b 3 b 2 a 3 a 3 b 1 b 3 a 1 a 1 b 2 b 1 a 2 displaystyle mathbf a times mathbf b a_ 2 b_ 3 b_ 2 a_ 3 a_ 3 b_ 1 b_ 3 a_ 1 a_ 1 b_ 2 b_ 1 a_ 2 and can also be written in components using einstein summation convention as a b i ε i j k a j b k displaystyle mathbf a times mathbf b _ i varepsilon _ ijk a_ j b_ k where ε i j k displaystyle varepsilon _ ijk is the levi civita symbol 44 it has the property that a b b a displaystyle mathbf a times mathbf b mathbf b times mathbf a 41 its magnitude is related to the angle θ displaystyle theta between a displaystyle mathbf a and b displaystyle mathbf b by the identity 41 a b a b sin θ displaystyle left mathbf a times mathbf b right left mathbf a right cdot left mathbf b right cdot left sin theta right the space and product form an algebra over a field which is not commutative nor associative but is a lie algebra with the cross product being the lie bracket 45 specifically the space together with the product r 3 displaystyle mathbb r 3 times is isomorphic to the lie algebra of three dimensional rotations denoted s o 3 displaystyle mathfrak so 3 43 in order to satisfy the axioms of a lie algebra instead of associativity the cross product satisfies the jacobi identity for any three vectors a b displaystyle mathbf a mathbf b and c displaystyle mathbf c 45 a b c b c a c a b 0 displaystyle mathbf a times mathbf b times mathbf c mathbf b times mathbf c times mathbf a mathbf c times mathbf a times mathbf b 0 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 46 abstract description edit see also vector space it can be useful to describe three dimensional space as a three dimensional vector space v displaystyle v over the real numbers this differs from r 3 displaystyle mathbb r 3 in a subtle way by definition there exists a basis b e 1 e 2 e 3 displaystyle mathcal b e_ 1 e_ 2 e_ 3 for v displaystyle v this corresponds to an isomorphism between v displaystyle v and r 3 displaystyle mathbb r 3 38 the construction for the isomorphism is found here however there is no preferred or canonical basis for v displaystyle v on the other hand there is a preferred basis for r 3 displaystyle mathbb r 3 which is due to its description as a cartesian product of copies of r displaystyle mathbb r that is r 3 r r r displaystyle mathbb r 3 mathbb r times mathbb r times mathbb r the three dimensional euclidean space 47 this allows the definition of canonical projections π i r 3 r displaystyle pi _ i mathbb r 3 rightarrow mathbb r where 1 i 3 displaystyle 1 leq i leq 3 for example π 1 x 1 x 2 x 3 x displaystyle pi _ 1 x_ 1 x_ 2 x_ 3 x this then allows the definition of the standard basis b standard e 1 e 2 e 3 displaystyle mathcal b _ text standard e_ 1 e_ 2 e_ 3 defined by π i e j δ i j displaystyle pi _ i e_ j delta _ ij where δ i j displaystyle delta _ ij is the kronecker delta written out in full the standard basis is 48 e 1 1 0 0 e 2 0 1 0 e 3 0 0 1 displaystyle e_ 1 begin pmatrix 1 0 0 end pmatrix e_ 2 begin pmatrix 0 1 0 end pmatrix e_ 3 begin pmatrix 0 0 1 end pmatrix therefore r 3 displaystyle mathbb r 3 can be viewed as the abstract vector space together with the additional structure of a choice of basis conversely v displaystyle v can be obtained by starting with r 3 displaystyle mathbb r 3 and forgetting the cartesian product structure or equivalently the standard choice of basis as opposed to a general vector space v displaystyle v the space r 3 displaystyle mathbb r 3 is sometimes referred to as a coordinate space 49 physically it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem for example in a problem with rotational symmetry working with the more concrete description of three dimensional space r 3 displaystyle mathbb r 3 assumes a choice of basis corresponding to a set of axes but in rotational symmetry there is no reason why one set of axes is preferred to say the same set of axes which has been rotated arbitrarily stated another way a preferred choice of axes breaks the rotational symmetry of physical space computationally it is necessary to work with the more concrete description r 3 displaystyle mathbb r 3 in order to do concrete computations affine description edit see also affine space and euclidean space ...
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