# Optimum Techniques for the Conversion of Space Rectangular and Curvilinear Coordinates

## Abstract

Conversion between space rectangular (X, Y, Z) and curvilinear (φ, λ, h) coordinates is an important task in the field of Surveying, geodesy, positioning, navigation, mapping etc. Different techniques which include iterative methods, non-iterative techniques and closed form algebraic methods have been applied over the years to carry out the coordinate conversion. However, the results obtained using these techniques are deficient in one way or the other due to the inherent limitations such as inability to produce results for curvilinear coordinates when the values of X, Y and Z are subsequently or simultaneously equal to zero. Therefore, this study attempts to put forth an optimum coordinate conversion technique between space rectangular and curvilinear coordinates. The data used are coordinates of points which include the space rectangular coordinates and their equivalent curvilinear coordinates. They were observed and processed in Nigeria using Doppler 9 software by African Doppler Survey (ADOS) and they were confirmed to be of first order accuracy and hence of high quality. The data processing involved the design of the optimum techniques equations, coding of the algorithms and necessary computations to obtain results. Analyzing the results obtained, it can be inferred that the designed optimum model has successfully carried out the conversion between space rectangular and curvilinear coordinates. Therefore, the optimum technique model is recommended for use for the conversions from Space rectangular coordinates to Geocentric, Geodetic, Reduced coordinates and vice versa.

Keywords: Coordinates Conversion, Curvilinear Coordinates, Optimum Techniques, Space Rectangular Coordinates

## References

W. E. Featherstone and P. Vaníček, “The role of coordinate systems, coordinates and heights in horizontal datum transformations”, The Australian Surveyor, 44, 143−150, 1999.

T. Fukushima, “Transformation from Spatial to Geographic coordinates Accelerated by Halley’s Method”,. Journal of Geodesy, 79 (12): 689-693, 2006.

W. Heiskanen and H. Moritz, Physical Geodesy, W.H. Freeman and Company, San Francisco, CA, 1967.

B. Holfman-Wellenhof and H. Moritz, Physical Geodesy, 2nd Edition, Springer Wien, New York, 2006.

T. O. Idowu, E. G. Ayodele and J. D. Dodo, “A comparison of Methods of Computing Geodetic Coordinates from Cartesian Coordinates”, National Journal of Space Research 6: 1-20, 2009.

G. Panou, “Cartesian to Geodetic coordinates conversion on an oblate spheroid using the bisection method”, Researchgate.net/publication, 2019.

R. H. Rapp, Geometric Geodesy Part I. Lecture Notes, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio, USA pp 24-26,1991.

H. Vermeille, “Direct transformation from geocentric to geodetic coordinates”, J. Geodesy, 76, 451-454, doi:10.1007/s00190-002-0273-6, 2002.

G. Suara, “Development of a computational tool for the optimum conversion of space rectangular and curvilinear coordinates”, M.Tech Dissertation, Department of Surveying & Geoinformatics, Federal University of Technology, Akure, Ondo Nigeria, 2019.