PROJECTIVE GEOMETRY

branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, of them that result from projection. Common examples of projections are the shadows cast by opaque objects, motion pictures, and maps of the Earth's surface. One of the stimuli for the development of projective geometry was the need to understand perspective in drawing and painting, in which a three-dimensional scene or object is rendered as if projected onto the plane of the picture. Rigorous study of the discipline was initiated by the French engineer Jean-Victor Poncelet in about 1820. Modern projective geometers emphasize the mathematical properties of objects that are preserved in the images, despite the distortion of lengths, angles, and shapes that generally occurs. Such properties include the straightness of lines and the incidence of points and lines; that is, if a point lies on a line in the object, the image of the point lies on the image of the line. The mathematical requirement of a projection is that every point in the object and the corresponding point in the image must lie on a straight line, the projection ray, that passes through the centre of projection. In descriptive geometry, the centre of projection is assumed to be infinitely distant from the object, so that all the projection rays are parallel; further, the rays are perpendicular to the projection plane, so that the projection is called orthogonal. In the pinhole camera, or camera obscura, the centre of projection is the pinhole; because the pinhole lies between the object and the image, the image is inverted. branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, of them that result from projection. Common examples of projections are the shadows cast by opaque objects, motion pictures, and maps of the Earth's surface. From the Greek period to the end of the 18th century, two momentous geometrical discoveries were made. Both concerned what was destined to be projective geometry. One was a theorem (stated in detail below) that was discovered and proved in 1639 by the French mathematician Girard Desargues. The other was a significant broadening of a theorem (first known to Pappus of Alexandria in the 4th century) by Desargues's fellow countryman Blaise Pascal, in 1640. Yet, until the discovery of the basic postulates of projective geometry by Jean-Victor Poncelet nearly 200 years later, Desargues's and Pascal's theorems seemed like parts of Euclidean geometry that were unlike other parts. It is significant that in their statements the only relation that mattered was that of the incidence of points and straight lines; distance, angle, congruency, or similitude played no part. (The use of the relation of incidence in the two theorems was the first indication that within the general discipline of geometry there are properties that depend on measurementthose involving distance, angle, and so forth were to be called metrical propertiesand others that do not, which became the characteristics of projective geometry.) Despite the simplicity of the statements of these two basic theorems, an unusual amount of care had to be taken in proving (and even formulating) the theorems as soon as due regard was paid to what happened when two or more lines referred to in the theorems, instead of intersecting, became parallel. The difficulties encountered in generalizing the theorems of Desargues and PappusPascal were overcome; and, in effect, projective geometry was founded when Poncelet postulated points at infinity, such that every straight line is extended with a point at infinity and every plane with a line at infinity, this new point (or line) being the same point for distinct parallel lines (or planes). All infinite elements of space were supposed to lie on the infinite plane of space. The plane so defined was called the projective plane, and, by logical extension to the next dimension, the corresponding space was called projective space. Afterward, the distinction between original and added elements (the infinite elements) faded out. Parallelism disappeared as a relation; the only relation still left from the variety of Euclid's relations (such as betweenness, order, congruence, incidence) was that of incidence (or lying on). After Poncelet's contribution, the major problem was to free projective geometry from its Euclidean substrate and to found it independently. This problem seemed to have been solved when Karl Georg Christian von Staudt, in Germany, started in 1847 with such incidence axioms as: two points determine one straight line; three points, not on a straight line, determine a plane; two planes intersect in a straight line, and so on. Though von Staudt's performance was admirable, he still took the relations of order (such as there being at least one point between two distinct points) in line and plane for granted without feeling the need for explicit formulations. Even worse, he in effect assumed that things proved for commensurable ratios (those expressible as rational numbers) were valid for all ratios, though centuries earlier Archimedes had provided an axiom granting the possibility of bracketing incommensurable ratios by larger and smaller commensurable ones. In 1873 Klein discovered the gap in von Staudt's reasoning. Though convinced of the possibility of founding projective geometry independently, Klein did not succeed in filling the gap; he did, however, make other essential contributions to the subject. A final solution for founding projective geometry independently required the following types of postulates: the incidence axioms of the type suggested above, axioms that order the points on the projective line in such a way that the order is cyclic (or as though on a circle) and axioms that preserve this order, and, finally, some type of Archimedes axiom of the kind mentioned in the previous paragraph. This solution was possible after Georg Cantor and Richard Dedekind, German innovators in mathematics, re-examined the ancient problem of incommensurables. Their main contributions in this direction were in 1871 and 1872. About 1880 Moritz Pasch of Germany and Otto Stolz of Austria recognized the import of the Archimedean axiom. Most European geometers of the 19th century made some contributions to the study of projective geometry in general. The axioms and logical structure of the subject have been studied by many mathematicians. Additional reading Studies of projective geometry include Dirk J. Struik, Lectures on Analytic and Projective Geometry (1953); H.S.M. Coxeter, The Real Projective Plane, 3rd ed. (1993), a meticulous but very readable exposition of the axiomatic approach to real projective geometry of two dimensions; A. Adrian Albert and Reuben Sandler, An Introduction to Finite Projective Planes (1968); C.W. O'Hara and D.R. Ward, An Introduction to Projective Geometry (1937, reprinted 1949); G.B. Mathews, Projective Geometry (1914), an easy and comparatively unsophisticated account for beginners; and Pierre Samuel, Projective Geometry (1988; originally published in French, 1986). Reinhold Baer, Linear Algebra and Projective Geometry (1952, reissued 1965); and A. Seidenberg, Lectures in Projective Geometry (1962), are more advanced texts. A.N. Whitehead, The Axioms of Projective Geometry (1906, reissued 1971); and A. Heyting, Axiomatic Projective Geometry, 2nd ed. (1980), are studies from the axiomatic approach. Luigi Cremona, Elements of Projective Geometry, 3rd ed. (1913, reprinted 1960), stands as a classic in the field. Oswald Veblen and John Wesley Young, Projective Geometry, 2 vol. (191018, reissued 1946), is the classic work in English on the axiomatic approach to projective geometry in two and three dimensions. Beniamino Segre, Lectures on Modern Geometry (1961), combines and contrasts the axiomatic and algebraic approaches to projective geometry, with a substantial algebraic introduction and a good deal of attention to finite, non-Pappian, and non-Desarguesian geometries. Lynn E. Garner, An Outline of Projective Geometry (1981), employs an abstract approach. Patrick Du Val The Editors of the Encyclopdia Britannica