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nn and giuseppe peano the latter of whom first gave the modern definition of vector spaces as an algebraic structure ref cite book title linear algebra a geometric approach first1 theodore last1 shifrin first2 malcolm last2 adams publisher w h freeman company year 2002 page 215 isbn 978 0 7167 4337 8 url https books google com books id e x8wgy9a5gc pg pa215 ref the development of matrix mathematics matrix mathematics and its application to n dimensional geometry was made by arthur cayley ref cite web title arthur cayley website mactutor first1 j j last1 o connor first2 e f last2 robertson date november 2014 publisher school of mathematics and statistics university of st andrews scotland url https mathshistory st andrews ac uk biographies cayley access date 2025 11 05 ref in euclidean geometry coordinate systems main coordinate system general geometry expanded 3d in 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 math x y and math z 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 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 ref name hughes cite book last1 hughes hallett first1 deborah author link deborah hughes hallett last2 mccallum first2 william g author2 link william g mccallum last3 gleason first3 andrew m author3 link andrew m gleason title calculus single and multivariable date 2013 publisher john wiley isbn 978 0470 88861 2 edition 6 ref 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 ref cite book title a student s guide to vectors and tensors series student s guides first daniel a last fleisch publisher cambridge university press year 2011 isbn 978 1 139 50394 5 pages 15 18 url https books google com books id eu1wcirdwsec pg pa15 ref ref cite book title mathematics for physical science and engineering symbolic computing applications in maple and mathematica first frank e last harris publisher academic press year 2014 isbn 978 0 12 801049 5 pages 202 205 url https books google com books id tbbragaaqbaj pg pa202 ref for more see euclidean space below are images of the above mentioned systems gallery image coord xyz svg cartesian coordinate system image cylindrical coordinates svg cylindrical coordinate system image spherical coordinates colatitude longitude svg spherical coordinate system gallery lines and planes two distinct points always determine a straight line geometry line three distinct points are either collinear or determine a unique plane geometry plane on the other hand four distinct points can either be collinear coplanar or determine the entire space ref cite book title introduction to the geometry of n dimensions series dover books on mathematics first d m y last sommerville author link duncan sommerville edition reprint publisher courier dover publications orig year 1929 year 2020 isbn 978 0 486 84248 6 pages 3 6 url https books google com books id 4vxddwaaqbaj pg pa3 ref 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 ref name bronštejn_semendjaev_2013 cite book title handbook of mathematics first1 ilja n last1 bronštejn first2 konstantin a last2 semendjaev edition 3rd publisher springer year 2013 isbn 978 3 662 25651 0 page 177 url https books google com books id mpxxcaaaqbaj pg pa177 ref file relations between planes png right thumb 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 ref name bronštejn_semendjaev_2013 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 ref cite book title geometry series springer undergraduate mathematics series first roger last fenn publisher springer science business media year 2012 page 152 isbn 978 1 4471 0325 7 url https books google com books id b1hlbwaaqbaj pg pa152 ref a line can lie in a given plane intersect that plane in a unique point or be parallel to the plane ref name bronštejn_semendjaev_2013 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 ref cite book title topics in mathematical analysis and differential geometry series pure mathematics volume 24 first nicolas k last laos publisher world scientific year 1998 isbn 978 981 02 3180 4 pages 220 221 url https books google com books id 1r7dsn4zqogc pg pa220 ref varignon s theorem states that the midpoints of any quadrilateral in math mathbb r 3 math form a parallelogram and hence are coplanar ref cite book title geometry by its history series undergraduate texts in mathematics first1 alexander last1 ostermann first2 gerhard last2 wanner publisher springer science business media year 2012 isbn 978 3 642 29163 0 url https books google com books id eosqphwwjx8c pg pa264 ref spheres and balls main sphere file sphere wireframe 10deg 6r svg right thumb a 3d projection perspective projection perspective projection of a sphere onto two dimensions a sphere in 3 space also called a 2 sphere because like all surface mathematics surface s it is intrinsically two dimensional consists of the set of all points in 3 space at a fixed distance math r from a central point mvar p the solid enclosed by the sphere is called a ball mathematics ball or 3 ball ref cite book title geometry of lengths areas and volumes volume 108 first james w last cannon publisher american mathematical society year 2017 isbn 978 1 4704 3714 5 page 29 url https books google com books id ssi_dwaaqbaj pg pa29 ref the volume of the ball is given by ref name johnston wilder_mason_2005 cite book title developing thinking in geometry editor1 first sue editor1 last johnston wilder editor2 first john editor2 last mason publisher paul chapman educational publishing year 2005 isbn 978 1 4129 1169 6 page 106 url https books google com books id kig6nvsg9oic pg pa106 ref math display block v frac 4 3 pi r 3 math and the surface area of the sphere is ref name johnston wilder_mason_2005 math display block a 4 pi r 2 math 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 math r sup 4 sup if a point has coordinates math p x y z w then math 1 x sup 2 sup y sup 2 sup z sup 2 sup w sup 2 sup 1 characterizes those points on the unit 3 sphere centered at the origin ref cite web title hypersphere last weisstein first eric w work wolfram mathworld url https mathworld wolfram com hypersphere html access date 2025 11 06 archive url https web archive org web 20260202033726 https mathworld wolfram com hypersphere html archive date 2026 02 02 ref this 3 sphere is an example of a 3 manifold a topological space space which is looks locally like 3d space ref cite book title discrete and computational geometry first1 joseph last1 o rourke first2 satyan l last2 devadoss publisher princeton university press year 2011 isbn 978 1 4008 3898 1 url https books google com books id injl6iaaiqqc pg pa146 ref in precise topological terms each point of the 3 sphere has a neighborhood which is homeomorphic to an open subset of 3d space polytopes main polyhedron in three dimensions there are nine regular polytope s the five convex platonic solid s and the four nonconvex kepler poinsot polyhedra ref cite book title gems of geometry first john last barnes edition 2nd publisher springer science business media year 2012 isbn 978 3 642 30964 9 page 46 url https books google com books id 7ycubud 4bqc pg pa46 ref class wikitable regular polytopes in three dimensions align center class colspan 5 platonic solid s colspan 4 kepler poinsot polyhedra polyhedral symmetry symmetry tetrahedral symmetry t sub d sub colspan 2 octahedral symmetry o sub h sub colspan 6 icosahedral symmetry i sub h sub coxeter group a sub 3 sub 3 3 colspan 2 b sub 3 sub 4 3 colspan 6 h sub 3 sub 5 3 align center symmetry order order 24 colspan 2 48 colspan 6 120 align center regular polyhedron regular br polyhedron file tetrahedron svg 50px br tetrahedron 3 3 file hexahedron svg 50px br cube 4 3 file octahedron svg 50px br octahedron 3 4 file dodecahedron svg 50px br dodecahedron 5 3 file icosahedron svg 50px br icosahedron 3 5 file smallstellateddodecahedron jpg 50px br small stellated dodecahedron 5 2 5 file greatdodecahedron jpg 50px br great dodecahedron 5 5 2 file greatstellateddodecahedron jpg 50px br great stellated dodecahedron 5 2 3 file greaticosahedron jpg 50px br great icosahedron 3 5 2 surfaces of revolution main surface of revolution a 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 ref name caliò_lazzari_2020 cite book title elements of mathematics with numerical applications first1 franca last1 caliò first2 alessandro last2 lazzari publisher società editrice esculapio year 2020 isbn 978 88 358 1755 0 pages 149 151 url https books google com books id bx_gdwaaqbaj pg pa149 ref ref name parker_1987 cite book title sceno graphic techniques first wilford oren last parker publisher siu press year 1987 isbn 978 0 8093 1350 1 pages 74 76 url https books google com books id be14qtt8sl4c pg pa74 ref 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 ref name caliò_lazzari_2020 ref name parker_1987 quadric surfaces main quadric surface in analogy with the conic section s the set of points whose cartesian coordinates satisfy the general equation of the second degree namely math display block ax 2 by 2 cz 2 fxy gyz hxz jx ky lz m 0 math where math a b c f g h j k l and math m are real numbers and not all of math a b c f g and math h are zero is called a quadric surface ref name brannan_et_al_2011 cite book title geometry display authors 1 first1 david a last1 brannan first2 matthew f last2 esplen first3 jeremy j last3 gray edition 2nd revised publisher cambridge university press year 2011 isbn 978 1 139 50370 9 pages 42 43 48 52 url https books google com books id ulrmkjijrzqc pg pa42 ref there are six types of degeneracy mathematics non degenerate quadric surfaces ref name brannan_et_al_2011 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 pi and all the lines of math r sup 3 sup through that conic that are normal to pi ref name brannan_et_al_2011 elliptic cones are sometimes considered to be degenerate quadric surfaces as well cn date november 2025 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 of one family intersects with just one exception every member of the other family ref name brannan_et_al_2011 each family is called a regulus geometry regulus ref cite book title the collected mathematical papers of arthur cayley volume 11 first arthur last cayley publisher cambridge university press year 1896 page 633 url https books google com books id o66akmm1a4c pg pa633 ref in linear algebra 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 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 vector s ref name towers_1988 cite book title guide to linear algebra series mathematical guides first david a last towers publisher bloomsbury publishing year 1988 isbn 978 1 349 09318 2 pages 6 8 url https books google com books id zzbkeaaaqbaj pg pa8 ref dot product angle and length main 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 math mathbb r 3 math 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 math 1 a a sub 1 sub a sub 2 sub a sub 3 sub and math 1 b b sub 1 sub b sub 2 sub b sub 3 sub is defined as ref name williams_2007 cite book title linear algebra with applications first gareth last williams publisher jones bartlett publishing year 2007 isbn 978 0 7637 5753 3 pages 38 40 url https books google com books id hlq9ocwuczmc pg pa38 ref math mathbf a cdot mathbf b a_1b_1 a_2b_2 a_3b_3 sum_ i 1 3 a_i b_i math the magnitude of a vector math a is denoted by math a the dot product of a vector math 1 a a sub 1 sub a sub 2 sub a sub 3 sub with itself is math mathbf a cdot mathbf a mathbf a 2 a_1 2 a_2 2 a_3 2 math which gives ref name williams_2007 math mathbf a sqrt mathbf a cdot mathbf a sqrt a_1 2 a_2 2 a_3 2 math 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 math a and math b is given by ref name williams_200...