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

Pędzich Paweł

Polish Cartographical Review

2022

Vol. 54, No. 1

35--53

The paper presents a review of the conformal projections of a tri-axial ellipsoid and the methodology of creating these projections with the use of isometric coordinates. The concept is very simple and has been known for a long time; if isometric coordinates are introduced on the surface of the original and on the plane of the image, then any analytical function of the complex variable, i.e. a function that has a continuous derivative, creates a conformal projection. The introduction presents the history of conformal projections. Then, existing projections are presented, including the Bugayevskiy projection and several projections developed by the author that apply selected functions of the complex variable. Scripts were prepared in the Octave software with the use of the presented methodology. Programming in Octave offers a possibility of a simple implementation of complex variable functions, which is also briefly discussed in the paper. The developed scripts were then used to perform calculations and to draw cartographic grids and distortion isolines in the selected conformal projections. The test object was the tri-axial ellipsoid that represents Phobos.

O pewnej numerycznej metodzie generowania współczynników w odwzorowaniach konforemnych

Pędzich P.

Roczniki Geomatyki

2007

T. 5, z. 3

133-145

In the paper, a method of computing polynomial coefficients approximating conformal map projections is presented. This method may be applied to creation of conformal projections of the ellipsoidal areas satisfying the criteria determining in detail the way of projection of selected parametric lines (meridians or parallels of latitude). The method is based on numerical solution of Euler-Urmayev differential equations. Some examples of this method are given. Also, the paper contains results of calculation of coordinates and local scales of linear distortions in selected projections with the use of the method described in the paper and of analytical formulas generally used in cartography.