Inmathematics,affine geometry is what remains ofEuclidean geometry when ignoring (mathematicians often say "forgetting"[1][2]) themetric notions ofdistance andangle.
As the notion ofparallel 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 lineL and a pointP not onL, there is exactly one line parallel toL that passes throughP.) 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 space is a set ofpoints to 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 space is a set ofpoints equipped with a set oftransformations (that isbijective mappings), thetranslations, which forms avector space (over a givenfield, commonly thereal numbers), and such that for any givenordered pair of points there is a unique translation sending the first point to the second; thecomposition 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 inone-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 ofmanifolds. That is developed in the articleAffine connection.
In 1748,Leonhard Euler introduced the termaffine[4][5] (from Latin affinis 'related') in his bookIntroductio in analysin infinitorum (volume 2, chapter XVIII). In 1827,August Möbius wrote 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 Weyl referred 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. Whittaker wrote:[8]
Several axiomatic approaches to affine geometry have been put forward:
As affine geometry deals with parallel lines, one of the properties of parallels noted byPappus of Alexandria has been taken as a premise:[9][10]
The full axiom system proposed haspoint,line, andline containing point asprimitive notions:
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 geometry but also inMinkowski'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 forrotation. Euclidean geometry corresponds to theordinary idea of rotation, while Minkowski's geometry corresponds tohyperbolic rotation. With respect toperpendicular lines, they remain perpendicular when the plane is subjected to ordinary rotation. In the Minkowski geometry, lines that arehyperbolic-orthogonal remain in that relation when the plane is subjected to hyperbolic rotation.
An axiomatic treatment of plane affine geometry can be built from theaxioms of ordered geometry by the addition of two additional axioms:[12]
The affine concept of parallelism forms anequivalence 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.
The firstnon-Desarguesian plane was noted byDavid Hilbert in hisFoundations of Geometry.[13] TheMoulton plane is a standard illustration. In order to provide a context for such geometry as well as those whereDesargues theorem is 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 relation between "vectors" defined by pairs of points from the plane.[14] Furthermore, the vectors form anabelian group underaddition; the ternary ring is linear and satisfiesright distributivity:
Geometrically, affine transformations (affinities) preservecollinearity: so they transform parallel lines into parallel lines and preserveratios of distances along parallel lines.
We identify asaffine theorems any geometric result that isinvariant under theaffine group (inFelix Klein'sErlangen programme this is its underlyinggroup of symmetry transformations for affine geometry). Consider in a vector spaceV, thegeneral linear groupGL(V). It is not the wholeaffine group because we must allow alsotranslations by vectorsv inV. (Such a translation maps anyw inV tow +v.) The affine group is generated by the general linear group and the translations and is in fact theirsemidirect product (Here we think ofV as a group under its operation of addition, and use the definingrepresentation ofGL(V) onV to define the semidirect product.)
For example, the theorem from the plane geometry of triangles about the concurrence of the lines joining eachvertex to themidpoint of the opposite side (at thecentroid orbarycenter) depends on the notions ofmid-point andcentroid as affine invariants. Other examples include the theorems ofCeva andMenelaus.
Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form anenvelope 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 unitisosceles right angled triangle to give 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 thevolume of 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 cube formed 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 thecenter of the base, and those with base aparallelogram instead of a square. The formula further generalizes to pyramids whose base can be dissected into parallelograms, includingcones by allowing infinitely many parallelograms (with due attention to convergence). The same approach shows that a four-dimensional pyramid has4Dhypervolume one quarter the3D volume of itsparallelepiped base times theheight, and so on for higher dimensions.
Two types of affine transformation are used inkinematics, both classical and modern.Velocityv 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 mapping of a plane with an axis for each represents coordinate change for an observer moving with velocityv 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 involvesrapidity instead of velocity, and substitutessqueeze mapping for the shear mapping used earlier. This affine geometry was developedsynthetically in 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 Nomizu in an article entitled "Trigonometry in Lorentzian geometry".[18]
Affine geometry can be viewed as the geometry of anaffine space of a given dimensionn, coordinatized over afieldK. There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed insyntheticfinite geometry. In projective geometry,affine space means the complement of ahyperplane at infinity in aprojective 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 example2x −y,x −y +z,(x +y +z)/3,ix + (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 asconfigurations 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, 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.
In traditionalgeometry, affine geometry is considered to be a study betweenEuclidean geometry andprojective geometry. On the one hand, affine geometry is Euclidean geometry withcongruence 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 thepoints at infinity.[19] In affine geometry, there is nometric structure but theparallel postulate does hold. Affine geometry provides the basis for Euclidean structure whenperpendicular lines are defined, or the basis for Minkowski geometry through the notion ofhyperbolic orthogonality.[20] In this viewpoint, anaffine transformation is aprojective transformation that does not permute finite points with points at infinity, and affinetransformation geometry is the study of geometrical properties through theaction of thegroup of affine transformations.
Abstract Algebra/Shear and Slope at Wikibooks
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