Hypergeodesic mapping of a surface on a plane

Abstract

The purpose of this paper is to study a projective analogue of the metric problemt of Beltrami, to map a surface of ordinary space on a plane so that the geodesics on the surface correspond to the straight lines in the plane. A two-parameter family of hypergeodesics on a surface in ordinary space is defined$ by a differential equation of the same form as the equation of the geodesics, in which, however, the coefficients are arbitrary given functions of the parameters on the surface instead of being the well known formulas containing the Christoffel symbols that appear in the familiar equation of the geodesics. The problem under consideration, then, is to set up a one-toone correspondence between the points of the surface and the points of a plane so that a given covariant family of hypergeodesics shall correspond to the straight lines of the plane. In the metric theory it is customary not only to suppose that two corresponding points on the surface and in the plane have the same curvilinear co6rdinates, but also that these are the abscissa and ordinate of the point in the plane. In the projective theory here presented it is found to be convenient first to normalize the curvilinear coordinates on the surface according to Fubini, and then to suppose that the curvilinear coordinates in the plane are the same as on the surface, so that Wilczynski's theory of plane nets is employed in studying the correspondence. In ?2, the theory of Fubini's canonical form of the differential equations of a surface is briefly summarized in so far as it will be needed in this paper. In ?3, the theory of hypergeodesics, including three examples, is similarly treated; and likewise in ?4, Wilczynski's theory of plane nets. The theory of hypergeodesic mapping is constructed in ?5. Three examples of this mapping are discussed in ?6. Green's congruentially asso-