Affine geometry is a form of geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths cannot be compared in different directions (that is, Euclid's third and fourth postulates are meaningless). Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject First identified by Euler, many affine properties are familiar from Euclidean geometry, but also apply in Minkowski space. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Those properties from Euclidean geometry that are preserved by parallel projection from one plane to another are affine. In effect, affine geometry is a generalization of Euclidean geometry characterized by slant and scale distortions. Projective geometry is more general than affine since it can be derived from projective space by "specializing" any one plane. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. [1]
In the language of Klein's Erlangen program, the underlying symmetry in affine geometry is the group of affinities, that is, the group of transformations of which preserve collinearity. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector
Affine geometry can be developed in terms of the geometry of vectors, with or without the notion of coordinates. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added An affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0 (sometimes known as free vectors). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Thus, affine geometry can be seen as part of linear algebra. Linear algebra is the branch of Mathematics concerned with
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History
Euler coined[2] the word affine (from the German, affin). Only after Felix Klein's Erlangen program was affine geometry recognized for being a generalization of Euclidean geometry. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. [3]
Axioms for affine geometry
An axiomatic treatment of affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms. Ordered geometry is a form of Geometry featuring the concept of intermediacy (or "betweenness" but like Projective geometry, omitting the basic notion
- (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. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive
- (Desargues) Give seven distinct points A, A', B, B', C, C', O, such that AA', BB', and 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'. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles
The affine concept of parallelism forms an equivalence relation on lines. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"
Affine transformations
Geometrically, affine transformations (affinities) preserve collinearity. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector So they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines. Affinities only admit two types of isometry: half-turns and translations. For the Mechanical engineering and Architecture usage see Isometric projection. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction Both half-turns and translations are types of dilatations or homothecy. In Mathematics, a homothety (or homothecy or dilation) is a transformation of space which takes each line into a parallel line (in essence a
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). In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Consider in a vector space V, the general linear group GL(V). In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation It is not the whole affine group because we must allow also translations by vectors v in V. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction (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 
. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can (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 mid-point of the opposite side (at the centroid or barycentre). In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes The idea of mid-point is an affine invariant. Other examples include the theorems of Ceva, Menelaus. Ceva's theorem is a well-known theorem in elementary Geometry. Menelaus' theorem, attributed to Menelaus of Alexandria, is a theorem about Triangles in Plane geometry.
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. In Mathematics, an envelope of a family of Manifolds (especially a family of Curves is a manifold that is Tangent to each member of 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.
Affine space
Affine geometry can be viewed as the geometry of affine space, of a given dimension n, coordinatized over a field K. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry. A finite geometry is any geometric system that has only a finite number of points. In projective geometry, affine space means the complement of the points (the hyperplane) at infinity (see also projective space). A hyperplane is a concept in Geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which Affine space can also be viewed as a vector space with the subtraction and scalar multiplication operations. That is one precise way in which to 'forget the origin'.
Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines (or sometimes, in higher dimensions, hyperplanes). Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects
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. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or
Applications and relationships
The notions of affine geometry have applications, for example in differential geometry. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Given the close relation with linear algebra, applications are plentiful. Linear algebra is the branch of Mathematics concerned with
See also
References
- ^ Coxeter, H. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. Ordered geometry is a form of Geometry featuring the concept of intermediacy (or "betweenness" but like Projective geometry, omitting the basic notion Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons, p. 261. ISBN 0471504580.
- ^ Blaschke, Wilhelm (1954). Analytische Geometrie. Basel: Birkhauser, p. 31.
- ^ Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons, p. 191. ISBN 0471504580.
External links
- Basics of Affine Geometry (typeset in PDF format)
Dictionary
affine geometry
-noun
- (geometry) The branch of geometry dealing with affine transformations.
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