Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations. The name affine geometry, like projective geometry and Euclidean geometry, follows naturally from the Erlangen program of Felix Klein.

Affine geometry is a form of geometry featuring the unique parallel line property (see the parallel postulate) where the notion of angle is undefined and lengths cannot be compared in different directions (that is, Euclid's third and fourth postulates are ignored). 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.

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 generated by the linear transformations of a vector space together with the translations by a vector.

Affine geometry can be developed on the basis of linear algebra. One can define an affine space as a set of points equipped with a set of transformations, the translations, which forms (the additive group of) a vector space (over a given field), and such that for any given ordered pair of points there is a unique translation sending the first point to the second. In more concrete terms, this amounts to having an operation that associates to any two points a vector, another that allows translation of a point by a vector to give another point, which operations verify a number of axioms (notably 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 this approach can be characterized as obtaining the affine space from its associated vector space by "forgetting" the origin (zero vector).

History

In 1748 Euler introduced the term affine (Latin affinis, "related") in his book Introductio in analysin infinitorum (see chapter XVII). In 1827 August Möbius wrote on affine geometry in his Der barycentrische Calcul, chapter 3.

Only after Felix Klein's Erlangen program was affine geometry recognized for being a generalization of Euclidean geometry.

Systems of axioms

Several axiomatic approaches to affine geometry have been put forward:

Pappus' law

As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise:

• If $A, B, C$ are on one line and $A', B', C'$ on another, then

$(AB' \parallel A'B \ \and \ BC' \parallel B'C) \Rightarrow CA' \parallel C'A.$

The full axiom system proposed has point, line, and line containing point as primitive 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 be parallel to l.
• Every line contains at least two points.
• There are at least three points not belonging to one line.

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’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

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.

Ordered structure

An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms.

1. (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.
2. (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. 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 fields

In 1984 Wanda Szmielew published a fundamental study of affine systems. As an algebraic preliminary, axioms are stated for several algebraic structures from loops to fields. Ternary fields are introduced as a ternary operation $T(u, x, v)$ that satisfies nine axioms that make it behave like $u x + v,$ the archetype of an affine transformation of x. Ternary fields are also characterized as strong quasifields. Szmielew considers Desarguean as well as Pappian affine plane in the third chapter of From affine to Euclidean geometry.

Affine transformations

Main article: Affine transformation

Geometrically, affine transformations (affinities) preserve collinearity. So they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines.

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 $V \rtimes \mathrm{GL}(V)$. (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 $\tfrac{3}{4} \log_e(2) - \tfrac{1}{2},$ i.e. 0.019860... or less than 2%, for all triangles.

Familiar formulas such as half the base times the height for 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

Main article: 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 whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x−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 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.

Projective view

In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out, and on the other hand affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity. In affine geometry there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality. In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity

See also

• Non-Euclidean geometry
• Affine
• Ordered geometry
• Euclidean geometry

References

• Emil Artin (1957) Geometric Algebra, chapter 2: "Affine and projective geometry", Interscience Publishers.
• V.G. Ashkinuse & Isaak Yaglom (1962) Ideas and Methods of Affine and Projective Geometry (in Russian), Ministry of Education, Moscow.
• H. S. M. Coxeter (1955) "The Affine Plane", Scripta Mathematica 21:5–14, a lecture delivered before the Forum of the Society of Friends of Scripta Mathematica on Monday, April 26, 1954.
• Felix Klein (1939) Elementary Mathematics from an Advanced Standpoint: Geometry, translated by E. R. Hedrick and C. A. Noble, pp 70–86, Macmillan Company.
• Wanda Szmielew (1984) From Affine to Euclidean Geometry: an axiomatic approach, D. Reidel, ISBN 90-277-1243-3 .
• Oswald Veblen (1918) Projective Geometry, volume 2, chapter 3: Affine group in the plane, pp 70 to 118, Ginn & Company.

External links

• Jean H. Gallier (2001) Geometric Methods and Applications for Computer Science and Engineering, Chapter 2:Basics of Affine Geometry, Springer Texts in Applied Mathematics #38, chapter online from University of Pennsylvania (PDF).