Construction and computation of geometries on the ellipsoid

Abstract

This article describes how to construct a wide range of geometry objects (called GeographicGeometry objects) in the coordinate system of an ellipsoid such as the Geographic coordinate system. Each construction process is formulated analytically and algorithmically using a combination of a set of fairly well-known mathematical methods such as ellipsoid geodesic construction functions, spherical trigonometry and iterative refinement methods. Each such geometry object may efficiently be converted to a corresponding Cartesian geometry object in any map projection coordinate system using an approximation algorithm. This property makes them particularly useful as a coordinate-system-independent geometry representation. A geographic geometry object is normally topologically equivalent to its Cartesian geometry counterpart except for some discontinuity and singularity cases.

Geographical area selection and construction of a corresponding routing grid used for in-flight management system flight trajectory optimization

“…Second, it must include the direct trajectory, which corresponds to the minimal ground distance between the trajectory's departure and destination airports which ensures that the points of the trajectory presenting the minimal total ground distance will be considered in the subsequent trajectory optimization. Given that all trajectories are defined with reference to the earth's surface, which is an oblate ellipsoid, 20,22,23,[34][35][36] a direct trajectory corresponds to the orthodrome between the two delimiting points. [20][21][22][23][24][25][26][27] Third, given that an excessive increase in the total ground distance may eliminate and even reverse any gains due to favorable wind conditions, the selected area must ensure a specified maximal total ground distance.…”

“…The atmospheric predictions are generally provided by national agencies, such as Environment Canada, 32 as a set of GRIdded Binary (GRIB) files, 33 and may refer to a regional or global forecast. Each GRIB file describes the values of an atmospheric parameter at a set of points corresponding to a grid constructed following a specified type of cartographic projection, 21,23,34,35 metric, and resolution, for a specified altitude and time. Subsequently, an atmospheric parameter’s value corresponding to any location, altitude and instant of time is computed through linear interpolation.…”

“…He addressed this inefficiency by selecting a hexagonal-shaped geographical area. A third type of routing grid was considered in the optimization methods investigated at LARCASE, [9][10][11][12] constructed using a set of tracks: the orthodrome (also known as the geodesic) corresponding to the shortest distance between the initial and final waypoints, [20][21][22][23][24][25][26][27] and a number of 4-11 equally spaced, parallel tracks situated on both sides of the orthodrome. The grid points were uniformly distributed along the set of tracks.…”