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In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting"See also forgetful functor.) the metric space 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 and a point not on , there is exactly one line parallel to that passes through .) 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. (Reprint of the 1957 original; A Wiley-Interscience Publication)
In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some (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 ), 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 . That is developed in the article on the affine connection.
After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.
In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtractionHermann 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. . See Chapter 1 §2 Foundations of Affine Geometry, pp 16–27 at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:E. T. Whittaker (1958). From Euclid to Eddington: a study of conceptions of the external world, Dover Publications, p. 130.
The full axiom system proposed has point, line, and line containing point as :
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 space 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.Coxeter 1955, Affine plane, p. 8
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.
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.
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.Rafael Artzy (1965). Linear Geometry, Addison-Wesley, p. 213. Furthermore, the vectors form an abelian group under additive group; the ternary ring is linear and satisfies right distributivity:
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 , the general linear group . It is not the whole affine group because we must allow also translations by vectors in . (Such a translation maps any in to .) The affine group is generated by the general linear group and the translations and is in fact their semidirect product (Here we think of as a group under its operation of addition, and use the defining representation of on 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 theorem.
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 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 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.
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.Edwin B. Wilson & Gilbert N. Lewis (1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences 48:387–507 Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite to express the special theory of relativity. In 1984, "the affine plane associated to the Lorentz space" was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry".Graciela S. Birman & Katsumi Nomizu (1984). "Trigonometry in Lorentzian geometry", American Mathematical Monthly 91(9):543–9, Lorentzian affine plane: p. 544
Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines (or sometimes, in higher dimensions, ). 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.
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