Projective range

rdf:langString Projective range

rdf:langString In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: Two fundamental one-dimensional forms such as point ranges, pencils of lines, or of planes are defined as projective, when their members are in one-to-one correspondence, and a harmonic set of one ... corresponds to a harmonic set of the other. ... If two one-dimensional forms are connected by a train of projections and intersections, harmonic elements will correspond to harmonic elements, and they are projective in the sense of Von Staudt.