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A systematic approach for solving the great circle track problems based on vector algebra

Chen C. L.

Polish Maritime Research

2016

nr 2

3--13

A systematic approach, based on multiple products of the vector algebra (S-VA), is proposed to derive the spherical triangle formulae for solving the great circle track (GCT) problems. Because the mathematical properties of the geometry and algebra are both embedded in the S-VA approach, derivations of the spherical triangle formulae become more understandable and more straightforward as compared with those approaches which use the complex linear combination of a vector basis. In addition, the S-VA approach can handle all given initial conditions for solving the GCT problems simpler, clearer and avoid redundant formulae existing in the conventional approaches. With the technique of transforming the Earth coordinates system of latitudes and longitudes into the Cartesian one and adopting the relative longitude concept, the concise governing equations of the S-VA approach can be easily and directly derived. Owing to the advantage of the S-VA approach, it makes the practical navigator quickly adjust to solve the GCT problems. Based on the S-VA approach, a program namely GCTPro_VA is developed for friendly use of the navigator. Several validation examples are provided to show the S-VA approach is simple and versatile to solve the GCT problems.

Vector algebra for Steep Slope Model analysis

Kolecka N.

Landform Analysis

2012

Vol. 21

17-25

Geographic Information Systems offer many algorithms that allow analysis of digital elevation models. They work with both GRID and TIN data, but they are limited to 2.5D models, where one planar (X,Y) position refers to only one vertical (Z) value. In mountainous regions, however, many steep, vertical and even overhung parts of rock walls and slopes occur. GRID and TIN models in a standard projection are not capable to deal with such a relief as they are not able to capture all complexity of steep slopes that can be observed from the terrestrial perspective. Such a perspective can be introduced into GIS via computer graphics software that allows 3D surface modelling by means of mesh, e.g. 3D triangular network. The paper presents a concept that implements 3D mesh in GIS and utilizes vector algebra to analyze such a surface. The idea is based on using normal vectors to compute slope and aspect of each triangle in a mesh. The computed values are saved as their attributes. Complete procedures are written in Python programming language and implemented into popular GIS software to work as a plug-in tool.