This chapter defines the mathematical spaces to which the geometrical quantities discussed in the previous chapter—scalars, vectors, and the metric—belong. Its goal is to go from the concept of a vector as an object whose components transform as Tⁱ → 𝓡ⱼ ⁱTj under a change of frame to the ‘intrinsic’ concept of a vector, T. These concepts are also generalized to ‘tensors’. The chapter also briefly remarks on how to deal with non-Cartesian coordinates. The velocity vector v is defined as a ‘free’ vector belonging to the vector space ε3 which subtends ε3. As such, it is not bound to the point P at which it is evaluated. It is, however, possible to attach it to that point and to interpret it as the tangent to the trajectory at P.
AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.
The term ‘density of dwellings' used by geographers is conceptually different to the term ‘probability density of dwellings' used by statisticians. In Cartesian coordinates the numerical values only differ by a scaling constant, but in polar coordinates this is not the case. To help clear up the resulting confusion, this paper attempts to show the relation between the two concepts.
Abstract
The cubic representation of the Burmester curve in Cartesian coordinates has certain disadvantages when automated searches are carried out. A parametric representation of the curve would be ideal. A systematic search could then be carried out by tracking points in a continuous fashion along the curve. In addition, solution rectification methods could be applied to determine the feasible segments, and the search could be limited to those portions only. This paper presents an alternative scheme for parametrizing the Burmester curves, as opposed to the complex number approach used by Chase et al. It uses the graphical method as its basis. The final scheme is not single valued, as it involves a parameter value as well a sign variable, but otherwise fulfils the requirements for an automatic search. It is an improvement on the cubic representation as it is double valued, rather than triple valued. The basic theory associated with the parametrization and the issues arising out of it are developed.
Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid