Affine geometry

Inmathematics,affine geometryis what remains ofEuclidean geometrywhen ignoring (mathematicians often say "forgetting"^[1]^[2]) themetricnotions ofdistanceandangle.

As the notion ofparallel linesis 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 withaffine 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]

Insynthetic geometry,anaffine spaceis a set ofpointsto which is associated a set of lines, which satisfy someaxioms(such as Playfair's axiom).

Affine geometry can also be developed on the basis oflinear algebra.In this context anaffine spaceis a set ofpointsequipped with a set oftransformations(that isbijective mappings), thetranslations,which forms avector space(over a givenfield,commonly thereal numbers), and such that for any givenordered pairof points there is a unique translation sending the first point to the second; thecompositionof 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 inone-to-one correspondencewith 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 ofmanifolds.That is developed in the article on theaffine connection.

History

In 1748,Leonhard Eulerintroduced the termaffine^[4]^[5](fromLatinaffinis'related') in his bookIntroductio in analysin infinitorum(volume 2, chapter XVIII). In 1827,August Möbiuswrote on affine geometry in hisDer barycentrische Calcul(chapter 3).

AfterFelix Klein'sErlangen program,affine geometry was recognized as a generalization ofEuclidean geometry.^[6]

In 1918,Hermann Weylreferred to affine geometry for his textSpace, Time, Matter.He used affine geometry to introduce vector addition and subtraction^[7]at the earliest stages of his development ofmathematical physics.Later,E. T. Whittakerwrote:^[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 ofparallel transport[...using]worldlinesof light-signals in four-dimensional space-time. A short element of one of these world-lines may be called anull-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

Several axiomatic approaches to affine geometry have been put forward:

Pappus' law

Pappus' 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 byPappus of Alexandriahas been taken as a premise:^[9]^[10]

Suppose $A, B, C$ are on one line and $A', B', C'$ on another. If the lines $AB'$ and $A'B$ are parallel and the lines $BC'$ and $B'C$ are parallel, then the lines $CA'$ and $C'A$ are parallel. (This is the affine version ofPappus's hexagon theorem).

The full axiom system proposed haspoint,line,andline containing pointasprimitive 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 beparallelto $L$ .
Every line contains at least two points.
There are at least three points not belonging to one line.

According toH. 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 inEuclidean geometrybut also inMinkowski's geometryof 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 forrotation.Euclidean geometry corresponds to theordinary idea of rotation,while Minkowski's geometry corresponds tohyperbolic rotation.With respect toperpendicularlines, they remain perpendicular when the plane is subjected to ordinary rotation. In the Minkowski geometry, lines that arehyperbolic-orthogonalremain in that relation when the plane is subjected to hyperbolic rotation.

Ordered structure

An axiomatic treatment of plane affine geometry can be built from theaxioms of ordered geometryby 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 $A, A', B, B', C, C', O$ ,such that $AA', BB', CC'$ are distinct lines through $O$ ,and $AB$ is parallel to $A'B'$ ,and $BC$ is parallel to $B'C'$ ,then $AC$ is parallel to $A'C'$ .

The affine concept of parallelism forms anequivalence relationon 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

The firstnon-Desarguesian planewas noted byDavid Hilbertin hisFoundations of Geometry.^[13]TheMoulton planeis a standard illustration. In order to provide a context for such geometry as well as those whereDesargues theoremis valid, the concept of a ternary ring was developed byMarshall 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 anequivalence relationbetween "vectors" defined by pairs of points from the plane.^[14]Furthermore, the vectors form anabelian groupunderaddition;the ternary ring is linear and satisfiesright distributivity:

(a+b)c=ac+bc.

Affine transformations

Geometrically, affine transformations (affinities) preservecollinearity:so they transform parallel lines into parallel lines and preserveratiosof distances along parallel lines.

We identify asaffine theoremsany geometric result that isinvariantunder theaffine group(inFelix Klein'sErlangen programmethis is its underlyinggroupof symmetry transformations for affine geometry). Consider in a vector space $V$ ,thegeneral linear group $GL(V)$ .It is not the wholeaffine groupbecause we must allow alsotranslationsby 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 theirsemidirect product $V\rtimes \mathrm {GL} (V).$ (Here we think of $V$ as a group under its operation of addition, and use the definingrepresentationof $GL(V)$ on $V$ to define the semidirect product.)

For example, the theorem from the plane geometry of triangles about the concurrence of the lines joining eachvertexto themidpointof the opposite side (at thecentroidorbarycenter) depends on the notions ofmid-pointandcentroidas affine invariants. Other examples include the theorems ofCevaandMenelaus.

Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form anenvelopeinside 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 unitisosceles right angled triangleto give ${\tfrac {3}{4}}\log _{e}(2)-{\tfrac {1}{2}},$ i.e. 0.019860... or less than 2%, for all triangles.

Familiar formulas such as half the base times the height for thearea of a triangle,or a third the base times the height for thevolumeof apyramid,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 theunit cubeformed by aface(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 thecenterof the base, and those with base aparallelograminstead of a square. The formula further generalizes to pyramids whose base can be dissected into parallelograms, includingconesby allowing infinitely many parallelograms (with due attention to convergence). The same approach shows that a four-dimensional pyramid has4D hypervolumeone quarter the3Dvolume of itsparallelepipedbase times theheight,and so on for higher dimensions.

Kinematics

Two types of affine transformation are used inkinematics,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 ofabsolute space and time.Theshear mappingof a plane with an axis for each represents coordinate change for an observer moving with velocity $v$ in a restingframe of reference.^[15]

Finite light speed,first noted by the delay in appearance of themoons of Jupiter,requires a modern kinematics. The method involvesrapidityinstead of velocity, and substitutessqueeze mappingfor the shear mapping used earlier. This affine geometry was developedsyntheticallyin 1912.^[16]^[17]to express thespecial theory of relativity. In 1984, "the affine plane associated to theLorentzian vector spaceL²"was described by Graciela Birman andKatsumi Nomizuin an article entitled "Trigonometry in Lorentzian geometry".^[18]

Affine space

Affine geometry can be viewed as the geometry of anaffine spaceof a given dimensionn,coordinatized over afieldK.There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed insynthetic finite geometry.In projective geometry,affine spacemeans the complement of ahyperplane at infinityin aprojective space.Affine spacecan 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 planesare 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 asconfigurationsof 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, ingroup theory,and incombinatorics.

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 tosymmetry.

Projective view

In traditionalgeometry,affine geometry is considered to be a study betweenEuclidean geometryandprojective geometry.On the one hand, affine geometry is Euclidean geometry withcongruenceleft out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent thepoints at infinity.^[19]In affine geometry, there is nometricstructure but theparallel postulatedoes hold. Affine geometry provides the basis for Euclidean structure whenperpendicularlines are defined, or the basis for Minkowski geometry through the notion ofhyperbolic orthogonality.^[20]In this viewpoint, anaffine transformationis aprojective transformationthat does not permute finite points with points at infinity, and affinetransformation geometryis the study of geometrical properties through theactionof thegroupof affine transformations.

References

^Berger, Marcel(1987),Geometry I,Berlin: Springer,ISBN 3-540-11658-3
^See alsoforgetful 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, 1922Space 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 Plane,§ 2: Affine geometry as an independent system
^Coxeter 1955,Affine plane,p. 8
^Coxeter,Introduction to Geometry,p. 192
^David Hilbert,1980 (1899).The Foundations of Geometry,2nd ed., Chicago: Open Court, weblink fromProject Gutenberg,p. 74.
^Rafael Artzy(1965).Linear Geometry,Addison-Wesley,p. 213.
^Abstract Algebra/Shear and Slopeat Wikibooks
^Edwin B. Wilson&Gilbert N. Lewis(1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of theAmerican Academy of Arts and Sciences48:387–507
^Synthetic Spacetime,a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived byWebCite
^Graciela S. Birman &Katsumi Nomizu(1984). "Trigonometry in Lorentzian geometry",American Mathematical Monthly91(9):543–9, Lorentzian affine plane: p. 544
^H. S. M. Coxeter (1942).Non-Euclidean Geometry,University of Toronto Press,pp. 18, 19.
^Coxeter 1942, p. 178

External links

Peter Cameron'sProjective and Affine GeometriesfromUniversity of London.
Jean H. Gallier(2001).Geometric Methods and Applications for Computer Science and Engineering,Chapter 2:"Basics of Affine Geometry"(PDF), Springer Texts in Applied Mathematics #38, chapter online fromUniversity of Pennsylvania.

[1] Berger, Marcel(1987),Geometry I,Berlin: Springer,ISBN 3-540-11658-3

[2] See alsoforgetful functor.

[3] 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)

[4] Miller, Jeff."Earliest Known Uses of Some of the Words of Mathematics (A)".

[5] Blaschke, Wilhelm (1954).Analytische Geometrie.Basel: Birkhauser. p. 31.

[6] Coxeter, H. S. M. (1969).Introduction to Geometry.New York: John Wiley & Sons. pp.191.ISBN 0-471-50458-0.

[7] Hermann Weyl(1918)Raum, Zeit, Materie.5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922Space Time Matter,Methuen, rept. 1952 Dover.ISBN 0-486-60267-2.See Chapter 1 §2 Foundations of Affine Geometry, pp 16–27

[8] E. T. Whittaker(1958).From Euclid to Eddington: a study of conceptions of the external world,Dover Publications,p. 130.

[9] Veblen 1918: p. 103 (figure), and p. 118 (exercise 3).

[10] Coxeter 1955,The Affine Plane,§ 2: Affine geometry as an independent system

[11] Coxeter 1955,Affine plane,p. 8

[12] Coxeter,Introduction to Geometry,p. 192

[13] David Hilbert,1980 (1899).The Foundations of Geometry,2nd ed., Chicago: Open Court, weblink fromProject Gutenberg,p. 74.

[14] Rafael Artzy(1965).Linear Geometry,Addison-Wesley,p. 213.

[15] Abstract Algebra/Shear and Slopeat Wikibooks

[16] Edwin B. Wilson&Gilbert N. Lewis(1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of theAmerican Academy of Arts and Sciences48:387–507

[17] Synthetic Spacetime,a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived byWebCite

[18] Graciela S. Birman &Katsumi Nomizu(1984). "Trigonometry in Lorentzian geometry",American Mathematical Monthly91(9):543–9, Lorentzian affine plane: p. 544

[19] H. S. M. Coxeter (1942).Non-Euclidean Geometry,University of Toronto Press,pp. 18, 19.

[20] Coxeter 1942, p. 178

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