Affine Geometry

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geometry

geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.

Types of Geometry

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

Their Relationship to Each Other

The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.

The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy (“betweenness”) but not that of measurement.

An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). Topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.

The Axiomatic Approach to Geometry

Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts. Euclid first defined his basic terms, such as point and line, then stated without proof certain axioms and postulates about them that seemed to be self-evident or obvious truths, and finally derived a number of statements (theorems) from the postulates by means of deductive logic. This axiomatic method has since been adopted not only throughout mathematics but in many other fields as well. The close examination of the axioms and postulates of Euclidean geometry during the 19th cent. resulted in the realization that the logical basis of geometry was not as firm as had previously been supposed. New axiom and postulate systems were developed by various mathematicians, notably David Hilbert (1899).

Bibliography

See H. G. Forder, The Foundations of Euclidean Geometry (1927); H. S. M. Coxeter, Introduction to Geometry (2d ed. 1969).

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Affine Geometry

a branch of geometry that studies those properties of figures in a plane or in space that are preserved under any affine transformation of the plane or space. An example of such a transformation is the transformation of similitude. The properties of a geometric figure which are preserved under any affine transformation are naturally called the affine invariants of this figure. A basic affine invariant is the simple ratio of three points M1, M2, and M3 which lie on a straight line. If x1x2, and x3 are the respective abscissas of these points, then the simple ratio is equal to (x2x1) / (x3x2). The affine invariants of any system consisting of n points (n ≥ 4) can be expressed in terms of simple ratios. Hence, in particular, it follows that the center of gravity of a geometric figure is preserved in affine transformations. In arbitrary affine transformations, parallel straight lines remain parallel. The methods and facts of affine geometry are widely used in different branches of natural science—mechanics, theoretical physics, and astronomy. For example, small deformations of a continuous medium, elastic in the first approximation, can be studied by the methods of affine geometry.

REFERENCES

Aleksandrov, P. S. Lektsii po analiticheskoi geometrii. Moscow, 1968.
Efimov, N. V. Vysshaia geometriia, 4th ed. Moscow, 1961.
E. G. POZNIAK