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kite circle diameter circumference area three dimensional volume cube cuboid cylinder pyramid sphere four other dimensional tesseract hypersphere geometers by name aida aryabhata ahmes alhazen apollonius archimedes atiyah baudhayana bolyai brahmagupta cartan coxeter descartes euclid euler gauss gromov hilbert 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 minggatu euler sakabe aida 1700s 1900s gauss lobachevsky bolyai riemann klein poincaré hilbert minkowski cartan veblen coxeter present day atiyah gromov v t e in mathematics affine geometry is what remains of euclidean geometry when ignoring mathematicians often say forgetting 1 2 the metric notions of distance and angle as the notion of parallel lines is one of the main properties that is independent of any metric affine geometry is often considered as the study of parallel lines therefore playfair s axiom given a line l and a point p not on l there is exactly one line parallel to l that passes through p is fundamental in affine geometry comparisons of figures in affine geometry are made with affine transformations which are mappings that preserve alignment of points and parallelism of lines affine geometry can be developed in two ways that are essentially equivalent 3 in synthetic geometry an affine space is a set of points to which is associated a set of lines which satisfy some axioms such as playfair s axiom affine geometry can also be developed on the basis of linear algebra in this context an affine space is a set of points equipped with a set of transformations that is bijective mappings the translations which forms a vector space over a given field commonly the real numbers and such that for any given ordered pair of points there is a unique translation sending the first point to the second the composition of two translations is their sum in the vector space of the translations in more concrete terms this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point these operations are required to satisfy a number of axioms notably that two successive translations have the effect of translation by the sum vector by choosing any point as origin the points are in one to one correspondence with the vectors but there is no preferred choice for the origin thus an affine space may be viewed as obtained from its associated vector space by forgetting the origin zero vector the idea of forgetting the metric can be applied in the theory of manifolds that is developed in the article on the affine connection contents 1 history 2 systems of axioms 2 1 pappus law 2 2 ordered structure 2 3 ternary rings 3 affine transformations 3 1 kinematics 4 affine space 5 projective view 6 see also 7 references 8 further reading 9 external links history edit in 1748 leonhard euler introduced the term affine 4 5 latin affinis related in his book introductio in analysin infinitorum volume 2 chapter xviii in 1827 august möbius wrote on affine geometry in his der barycentrische calcul chapter 3 after felix klein s erlangen program affine geometry was recognized as a generalization of euclidean geometry 6 in 1918 hermann weyl referred to affine geometry for his text space time matter he used affine geometry to introduce vector addition and subtraction 7 at the earliest stages of his development of mathematical physics later e t whittaker wrote 8 weyl s geometry is interesting historically as having been the first of the affine geometries to be worked out in detail it is based on a special type of parallel transport using worldlines of light signals in four dimensional space time a short element of one of these world lines may be called a null vector then the parallel transport in question is such that it carries any null vector at one point into the position of a null vector at a neighboring point systems of axioms edit several axiomatic approaches to affine geometry have been put forward pappus law edit pappus s law if the red lines are parallel and the blue lines are parallel then the dotted black lines must be parallel as affine geometry deals with parallel lines one of the properties of parallels noted by pappus of alexandria has been taken as a premise 9 10 suppose a b c displaystyle a b c are on one line and a b c displaystyle a b c on another if the lines a b displaystyle ab and a b displaystyle a b are parallel and the lines b c displaystyle bc and b c displaystyle b c are parallel then the lines c a displaystyle ca and c a displaystyle c a are parallel the full axiom system proposed has point line and line containing point as primitive notions two points are contained in just one line for any line l and any point p not on l there is just one line containing p and not containing any point of l this line is said to be parallel to l every line contains at least two points there are at least three points not belonging to one line according to h s m coxeter the interest of these five axioms is enhanced by the fact that they can be developed into a vast body of propositions holding not only in euclidean geometry but also in minkowski s geometry of time and space in the simple case of 1 1 dimensions whereas the special theory of relativity needs 1 3 the extension to either euclidean or minkowskian geometry is achieved by adding various further axioms of orthogonality etc 11 the various types of affine geometry correspond to what interpretation is taken for rotation euclidean geometry corresponds to the ordinary idea of rotation while minkowski s geometry corresponds to hyperbolic rotation with respect to perpendicular lines they remain perpendicular when the plane is subjected to ordinary rotation in the minkowski geometry lines that are hyperbolic orthogonal remain in that relation when the plane is subjected to hyperbolic rotation ordered structure edit an axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms 12 affine axiom of parallelism given a point a and a line r not through a there is at most one line through a which does not meet r desargues given seven distinct points a a b b c c o displaystyle a a b b c c o such that a a displaystyle aa b b displaystyle bb and c c displaystyle cc are distinct lines through o displaystyle o and a b displaystyle ab is parallel to a b displaystyle a b and b c displaystyle bc is parallel to b c displaystyle b c then a c displaystyle ac is parallel to a c displaystyle a c the affine concept of parallelism forms an equivalence relation on lines since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers ternary rings edit main article planar ternary ring the first non desarguesian plane was noted by david hilbert in his foundations of geometry 13 the moulton plane is a standard illustration in order to provide a context for such geometry as well as those where desargues theorem is valid the concept of a ternary ring was developed by marshall hall in this approach affine planes are constructed from ordered pairs taken from a ternary ring a plane is said to have the minor affine desargues property when two triangles in parallel perspective having two parallel sides must also have the third sides parallel if this property holds in the affine plane defined by a ternary ring then there is an equivalence relation between vectors defined by pairs of points from the plane 14 furthermore the vectors form an abelian group under addition the ternary ring is linear and satisfies right distributivity a b c ac bc affine transformations edit main article affine transformation geometrically affine transformations affinities preserve collinearity so they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines we identify as affine theorems any geometric result that is invariant under the affine group in felix klein s erlangen programme this is its underlying group of symmetry transformations for affine geometry consider in a vector space v the general linear group gl v it is not the whole affine group because we must allow also translations by vectors v in v such a translation maps any w in v to w v the affine group is generated by the general linear group and the translations and is in fact their semidirect product v g l v displaystyle v rtimes mathrm gl v here we think of v as a group under its operation of addition and use the defining representation of gl v on v to define the semidirect product for example the theorem from the plane geometry of triangles about the concurrence of the lines joining each vertex to the midpoint of the opposite side at the centroid or barycenter depends on the notions of mid point and centroid as affine invariants other examples include the theorems of ceva and menelaus affine invariants can also assist calculations for example the lines that divide the area of a triangle into two equal halves form an envelope inside the triangle the ratio of the area of the envelope to the area of the triangle is affine invariant and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give 3 4 log e 2 1 2 displaystyle tfrac 3 4 log _ e 2 tfrac 1 2 i e 0 019860 or less than 2 for all triangles familiar formulas such as half the base times the height for the area of a triangle or a third the base times the height for the volume of a pyramid are likewise affine invariants while the latter is less obvious than the former for the general case it is easily seen for the one sixth of the unit cube formed by a face area 1 and the midpoint of the cube height 1 2 hence it holds for all pyramids even slanting ones whose apex is not directly above the center of the base and those with base a parallelogram instead of a square the formula further generalizes to pyramids whose base can be dissected into parallelograms including cones by allowing infinitely many parallelograms with due attention to convergence the same approach shows that a four dimensional pyramid has 4d hypervolume one quarter the 3d volume of its parallelepiped base times the height and so on for higher dimensions kinematics edit two types of affine transformation are used in kinematics both classical and modern velocity v is described using length and direction where length is presumed unbounded this variety of kinematics styled as galilean or newtonian uses coordinates of absolute space and time the shear mapping of a plane with an axis for each represents coordinate change for an observer moving with velocity v in a resting frame of reference 15 finite light speed first noted by the delay in appearance of the moons of jupiter requires a modern kinematics the method involves rapidity instead of velocity and substitutes squeeze mapping for the shear mapping used earlier this affine geometry was developed synthetically in 1912 16 17 to express the special theory of relativity in 1984 the affine plane associated to the lorentzian vector space l 2 was described by graciela birman and katsumi nomizu in an article entitled trigonometry in lorentzian geometry 18 affine space edit main article affine space affine geometry can be viewed as the geometry of an affine space of a given dimension n coordinatized over a field k there is also in two dimensions a combinatorial generalization of coordinatized affine space as developed in synthetic finite geometry in projective geometry affine space means the complement of a hyperplane at infinity in a projective space affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one for example 2 x y x y z x y z 3 i x 1 i y etc synthetically affine planes are 2 dimensional affine geometries defined in terms of the relations between points and lines or sometimes in higher dimensions hyperplanes defining affine and projective geometries as configurations of points and lines or hyperplanes instead of using coordinates one gets examples with no coordinate fields a major property is that all such examples have dimension 2 finite examples in dimension 2 finite affine planes have been valuable in the study of configurations in infinite affine spaces in group theory and in combinatorics despite being less general than the configurational approach the other approaches discussed have been very successful in illuminating the parts of geometry that are related to symmetry projective view edit in traditional geometry affine geometry is considered to be a study between euclidean geometry and projective geometry on the one hand affine geometry is euclidean geometry with congruence left out on the other hand affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity 19 in affine geometry there is no metric structure but the parallel postulate does hold affine geometry provides the basis for euclidean structure when perpendicular lines are defined or the basis for minkowski geometry through the notion of hyperbolic orthogonality 20 in this viewpoint an affine transformation is a projective transformation that does not permute finite points with points at infinity and affine transformation geometry is the study of geometrical properties through the action of the group of affine transformations see also edit non euclidean geometry references edit berger marcel 1987 geometry i berlin springer isbn 3 540 11658 3 see also forgetful functor artin emil 1988 geometric algebra wiley classics library new york john wiley sons inc pp x 214 doi 10 1002 9781118164518 isbn 0 471 60839 4 mr 1009557 reprint of the 1957 original a wiley interscience publication miller jeff earliest known uses of some of the words of mathematics a blaschke wilhelm 1954 analytische geometrie basel birkhauser p 31 coxeter h s m 1969 introduction to geometry new york john wiley sons pp 191 isbn 0 471 50458 0 hermann weyl 1918 raum zeit materie 5 edns to 1922 ed with notes by jūrgen ehlers 1980 trans 4th edn henry brose 1922 space time matter methuen rept 1952 dover isbn 0 486 60267 2 see chapter 1 2 foundations of affine geometry pp 16 27 e t whittaker 1958 from euclid to eddington a study of conceptions of the external world dover publications p 130 veblen 1918 p 103 figure and p 118 exercise 3 coxeter 1955 the affine...