Land surface in gravity points classification by a complete system of curvatures

Abstract

A general theory of land surface in a gravitational field is being developed. The four classes of topographic variables are defined: local (class A) and global (class B) that have no sense without gravity, and local (class C) and global (class D) which are gravity invariant. A complete system of curvatures is introduced for a general situation of nonuniform gravity. The curvatures refer to classes A and C, the latter being subject of the differential geometry of surfaces, the former being subject of this work in a special section of mathematics concerned with surfaces in a vector field. The svstem of curvatures consists of 7 known ones of the classes A and C and 5 new curvatures of the class A (difference, horizontal excess, vertical excess, total ring, and total accumulation curvatures). Seven new theorems show in which way curvatures can reflect landforms and their ability to influence substance flows, and the relationship between them. Land surface in gravity-points classification is constructed based on signs of curvatures, which includes known curvature-based classifications as partial situations, and 12 main types (from total 48) are shown to form open subsets of the surface with equal probability to meet them for the land surface as indicated by a new statistical hypothesis. A central-point method for local variables calculation in uniform gravity approximation is described for a computer Digital Elevation Models treatment.