Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

Abstract

Abstract:The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the second kind and Jacobian elliptic functions. This solution allows to use common known and widely described in literature methods of solving such integrals and functions. The main advantage of this method is the fact that the calculations of x and y coordinates are practically based on a single algorithm that is required to solve the elliptic integral of the second kind. Equations are provided for three types of map projections: cylindrical, azimuthal and pseudocylindrical. These types of projections are often used in planetary cartography for presentation of entire and polar regions of extraterrestrial objects. The paper also contains equations for the calculation of the length of a meridian and a parallel of a triaxial ellipsoid in reduced coordinates. Moreover, graticules of three coordinates systems (planetographic, planetocentric and reduced) in developed map projections are presented. The basic properties of developed map projections are also described. The obtained map projections may be applied in planetary cartography in order to create maps of extraterrestrial objects.

Conformal projections of a tri-axial ellipsoid based on isometric coordinates: history, methodology, and examples

“…As the data for multiple extra-terrestrial bodies are often expressed in form of planetocentric coordinates, the knowledge of the methods for converting the coordinates between different systems might be useful. These methods were described, among others, in the works Bugayevskiy (1998) and Pędzich (2017), and the main definitions were also presented in the study by Pędzich and Latuszek (2014). Here, it is worth noting that the planetocentric latitude is the angle between the plane of the equator and the straight line passing through the centre of the ellipsoid and the given point, while the planetographic latitude is the angle between the plane of the equator and the normal to the ellipsoid at a given point.…”