Affine geometry

plane
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v t e

In

.

As the notion of

, 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]

In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).

Affine geometry can also be developed on the basis of

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 "

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 manifolds. That is developed in the article on the affine connection.

History

In 1748,

wrote on affine geometry in his Der barycentrische Calcul (chapter 3).

After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.^[6]

In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtraction^[7] at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:^[8]

Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of

worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a null-vector

; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point.

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:^[9][10]

• Suppose A, B, C are on one line and A', B', C' on another. If the lines AB' and A'B are parallel and the lines BC' and B'C are parallel, then the lines CA' and C'A are parallel. (This is the affine version of Pappus's hexagon theorem).

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

:

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.^[11]

The various types of affine geometry correspond to what interpretation is taken for rotation. Euclidean geometry corresponds to the

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:^[12]

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', 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 rings

The first

.

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

right distributivity

:

${\displaystyle (a+b)c=ac+bc.}$

Affine transformations

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

${\displaystyle V\rtimes \mathrm {GL} (V).}$ (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

.

Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form an

to give ${\displaystyle {\tfrac {3}{4}}\log _{e}(2)-{\tfrac {1}{2}},}$ i.e. 0.019860... or less than 2%, for all triangles.

Familiar formulas such as half the base times the height for the

, and so on for higher dimensions.

Kinematics

Two types of affine transformation are used in kinematics, both classical and modern. Velocity v is described using length and direction, where length is presumed unbounded. This variety of kinematics, styled as Galilean or Newtonian, uses coordinates of absolute space and time. The shear mapping of a plane with an axis for each represents coordinate change for an observer moving with velocity v in a resting frame of reference.^[15]

. In 1984, "the affine plane associated to the Lorentzian vector space L²" was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry".^[18]

Affine space

Affine geometry can be viewed as the geometry of an 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 a hyperplane at infinity in a 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,

.

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

of affine transformations.

See also

• Non-Euclidean geometry

References

Further reading

• Emil Artin (1957) Geometric Algebra, chapter 2: "Affine and projective geometry", via Internet Archive
• V.G. Ashkinuse & Isaak Yaglom (1962) Ideas and Methods of Affine and Projective Geometry (in Russian), Ministry of Education, Moscow.
• M. K. Bennett (1995) Affine and Projective Geometry,
  ISBN 0-471-11315-8
  .
• 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.
• Macmillan Company
  .
• Bruce E. Meserve (1955) Fundamental Concepts of Geometry, Chapter 5 Affine Geometry, pp 150–84, Addison-Wesley.
• Peter Scherk & Rolf Lingenberg (1975) Rudiments of Plane Affine Geometry, Mathematical Expositions #20, University of Toronto Press.
• Wanda Szmielew (1984) From Affine to Euclidean Geometry: an axiomatic approach,
  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

Wikimedia Commons has media related to Affine geometry.

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