Projective geometry

Abstract

Projective geometry investigates those properties of geometrical figures that are unaltered by projection. The impetus for these investigations was provided by the study of perspective in painting and architecture. Following the development of descriptive geometry, principally by Gaspard Monge (1746–1818), Victor Poncelet (1788–1867) gave a first outline of projective geometry in his ‘Traité des propriétés projective s des figures’. Analytical methods in projective geometry were introduced mainly by August Ferdinand Möbius (1790–1868) and Julius Plücker (1801–1868), while Jacob Steiner (1796–1863) and Christian von Staudt (1798–1867) perfected a development of projective geometry without these methods. The first beginnings of this synthetic approach are to be found in the work of Pappus (250–300? B. C), who introduced the cross-ratio, referring to a lost work of Apollonius of Perga (265–180 B. C.?). The connection between projective and Euclidean geometry was clarified by Felix Klein (1849–1925). He also introduced the idea of a geometry as the invariant theory of a certain group of mappings.