Affine geometry


Affine geometry

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). First identified by Euler, many affine properties are familiar from Euclidean geometry, but also apply in Minkowski space. 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. [cite book | last=Coxeter | first=H. S. M. | pages=p. 261 | title=Introduction to Geometry | location=New York | publisher=John Wiley & Sons | year=1969 | isbn=0471504580]

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.

Affine geometry can be developed in terms of the geometry of vectors, with or without the notion of coordinates. An affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin "0" (sometimes known as "free vectors"). Thus, affine geometry can be seen as part of linear algebra.

History

Euler coined [cite book | last=Blaschke | first=Wilhelm | pages=p. 31 | title=Analytische Geometrie | location=Basel | publisher=Birkhauser | year=1954 | isbn=] 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. [cite book | last=Coxeter | first=H. S. M. | pages=p. 191 | title=Introduction to Geometry | location=New York | publisher=John Wiley & Sons | year=1969 | isbn=0471504580]

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.

#(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', 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'.

The affine concept of parallelism forms an equivalence relation on lines.

Affine transformations

Geometrically, affine transformations (affinities) preserve collinearity. 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. Both half-turns and translations are types of dilatations or homothecy.

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 times 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 mid-point 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 over 4} log_e(2)-{1 over 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 volume one quarter the 3D volume of its parallelopiped base times the height, and so on for higher dimensions.

Affine space

Affine geometry can be viewed as the geometry of 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 the points (the hyperplane) at infinity (see also projective space). "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). 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.

Applications and relationships

The notions of affine geometry have applications, for example in differential geometry. Given the close relation with linear algebra, applications are plentiful.

ee also

* Non-Euclidean geometry
* Affine
* Ordered geometry
* Euclidean geometry
* Erlangen program

References

External links

* [http://www.cis.upenn.edu/~cis610/geombchap2.pdf Basics of Affine Geometry] (typeset in PDF format)


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