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

Pędzich Paweł
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Abstract

Abstract 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.
基于等距坐标的三轴椭球体的保形投影:历史、方法和实例
摘要本文综述了三轴椭球体的保形投影以及利用等距坐标生成保形投影的方法。这个概念非常简单,并且已经存在了很长时间;如果在原始图像的表面和图像的平面上引入等距坐标,则任何复变量的解析函数,即具有连续导数的函数,都会产生保形投影。引言部分介绍了保形投影的历史。然后,介绍了现有的投影,包括Bugayevskiy投影和作者开发的几种投影,这些投影应用了复变量的选定函数。使用所提出的方法在Octave软件中编写脚本。在Octave中编程提供了一种简单实现复杂变量函数的可能性,本文也简要讨论了这一点。然后使用开发的脚本进行计算,并在选定的保形投影中绘制地图网格和失真等值线。测试对象是代表火卫一的三轴椭球体。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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