Efficient transformation from Cartesian to geodetic coordinates

Highlights

• Efficient computation of geodetic coordinates is achieved using new methods.

• Sampson's distance rule from computer vision is extended and modified.

• A new, more efficient implementation of Bowring's method is also suggested.

• Bowring's method is the most efficient of all if the new implementation is used.

Abstract

The derivation of algorithms for the computation of geodetic coordinates from 3D Cartesian coordinates has been a very active field of research among geodesists for more than forty years. Many authors have sought the most efficient method, i.e. the method that provides the fastest computational speed, which nevertheless yields sufficient accuracy for practical applications. The problem is a special case of a more general mathematical problem that has also been studied by researchers in other fields. This paper investigates the applicability of methods by Sampson (1982, Computer graphics and image processing, 18: 97–108) and Uteshev and Goncharova (2018, Journal of Computational and Applied Mathematics, 328: 232–251) to the computation of geodetic coordinates. Both methods have been modified to make them more suitable for this particular problem. The methods are compared to several commonly used geodetic methods in terms of accuracy and computational efficiency. It is found that a simple modification improves the accuracy of the methods by ~3 orders of magnitude, and the modified method of Uteshev and Goncharova (2018) achieves an accuracy of <0.1 mm anywhere on the surface of the Earth. The methods are especially efficient in the computation of ellipsoidal height. As an additional result of this study, a new formulation of the well-known method by Bowring (1976, Survey Review, 23: 323–327) is derived, and it is shown to improve the computation speed of Bowring's method by ~12%–~27% compared to the conventional formulation.

Introduction

The transformation from 3D Cartesian coordinates $\left(X,Y,Z\right)$ to geodetic coordinates (geodetic latitude $\phi$, longitude $\lambda$, and ellipsoidal height $h$) is a classical problem in geodesy and its application is extremely common. While the computation of longitude is straightforward, the computation of geodetic latitude and ellipsoidal height is more complicated. Many different methods have been published in the geodetic literature. An overview of many of these methods can be found in (Featherstone and Claessens, 2008), and many more have been published since (e.g., Turner, 2009; Shu and Li, 2010; Civicioglu, 2012; Ligas, 2012; Soler et al., 2012; Zeng, 2013). Most methods focus on the computation of geodetic latitude, after which the ellipsoidal height can readily be found, but it is equally possible to solve for the ellipsoidal height first and geodetic latitude second.

Methods for the computation of geodetic coordinates from Cartesian coordinates can be divided into three categories: exact, iterative and approximate methods. Here we define an approximate method as any method that is neither exact nor uses a variable number of iterations. For example, Bowring's (1976) method is iterative, but when implemented such that only a single iteration is used (as is often the case), we consider it an approximate method.

An exact solution involves the solution of a quartic equation (fourth-order polynomial) (e.g. Paul, 1973; Borkowski, 1989; Vermeille, 2004, 2011), which inevitably leads to a computationally inefficient algorithm. Geodesists have put much effort into devising more efficient iterative or approximate methods. Some of the simplest and most efficient of these are the methods by Bowring (1976, 1985) and Fukushima (1999, 2006).

In other fields, similar problems have been tackled in parallel. For example, in the field of computer vision, a common problem is the estimation of conic sections through scattered data points. To estimate a best fitting ellipse (in the case that the conic section is an ellipse), an approximation of the distance between a point and the ellipse is required. A well-known algorithm for this problem is provided by Sampson (1982), and the approximate distance has become known as Sampson's distance. Meanwhile, mathematicians have worked on more general problems, such as computation of the shortest distance between a point and any degree 2 curve or manifold in ${\mathbb{R}}^n$. For example, Uteshev and Yashina (2015) provide a method for finding the distance between an ellipsoid and any first- or second-order manifold. Explicit exact and approximate formulas for the distance between a point and an ellipse are provided in Uteshev and Goncharova (2018).

The main aim of this paper is to investigate the applicability of approximate solutions by Sampson (1982) and Uteshev and Goncharova (2018), from outside of the geodetic literature, to the computation of geodetic coordinates on or near Earth. These methods are then compared to a selection of geodetic methods in terms of accuracy and computational efficiency. The focus is on simple and efficient (fast) algorithms for the computation of geodetic coordinates that are precise enough for any practical application on the Earth's surface or at flight altitude.

The geodetic transformation problem is briefly defined in section 2. In section 3, Sampson's and Uteshev's methods are outlined. It will be shown that these methods are not sufficiently accurate for geodetic applications, except for points very close to the reference ellipsoid. However, new modifications to these methods to make them more suited to the geodetic coordinate transformation are presented in section 4. In section 5, the geodetic methods of Bowring (1976, 1985), Pollard (2002), and Fukushima (2006) are outlined. The accuracy of the unmodified and modified methods of Sampson (1982) and Uteshev and Goncharova (2018) are compared to these geodetic methods in section 6, and in section 7 a comparison in terms of computational efficiency is provided. An important point is made about the variability in computational efficiency for different hardware, software and implementation. Finally, section 8 provides conclusions and recommendations.