Nine-parameter curvilinear transformation between datums
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ORIGINAL PAPER
Nine-parameter curvilinear transformation between datums Ahmad H. Alashaikh 1 & Hasan M. Bilani 1 & Saad A. Alshehri 2
Received: 14 January 2015 / Accepted: 23 November 2015 / Published online: 24 February 2016 # Saudi Society for Geosciences 2016
Abstract One major challenge, in geodesy, is the conversion of geodetic coordinates related to a geodetic datum to geodetic coordinates related to another datum, due to differences between datums in size and position. Essentially, there are two approaches, where the first, and most well known, is conversion via three-dimensional Cartesian coordinate space. The second, curvilinear transformation, is a direct transformation of latitude-longitude coordinates between two datums. The method proposed in this paper is named nine-parameter curvilinear transformation between datums and may be considered a new curvilinear transformation method. This method has good accuracy, since the difference between suggested transformation and seven-parameter transformation does not exceed 0.0015 arc sec. Moreover, it replaces several steps in seven-parameter transformation methods by one direct step. Keywords Transformation . Conversion . Datum . Ellipsoid
Introduction The conversion of geodetic coordinates (B1, L1) related to geodetic datum 1 to (B2, L2) related to datum 2 presents a challenge due to rotations by very small angles α, β and γ respectively, and displacements ΔX, ΔY and ΔZ applied to
* Hasan M. Bilani [email protected] 1
Civil Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
2
Centre for Geospatial Science, King Fahd Security College, P. O. Box 377246, Riyadh 11335, Saudi Arabia
axes X, Y and Z, respectively, in the positive direction. Figure (1) shows the geometry of the two systems. Essentially, there are two types of conversion, one via three-dimensional Cartesian coordinate space, such as threeparameter, seven-parameter and seven + three parameter (Molodensky-Badekas) conversions. Second, curvilinear transformation, which is a direct transformation of latitudelongitude coordinates between two datums, such as geographic offsets, standard Molodensky, abridged Molodensky and multiple regression. The curvilinear transformations are conceptually more simple as they directly produce a coordinate change in degrees instead of converting via Cartesian coordinates (Featherstone 1997). In the three-dimensional Cartesian coordinate space methods, coordinates in datum 1 are converted from latitude-longitude into three-dimensional Cartesian coordinate. Next, shifts and rotations are applied to axes X, Y and Z. Finally, the coordinates are translated back into the latitudelongitude values of datum 2. Essentially, there are three three-dimensional transformation methods. The first is three-parameter transformation, also called a geocentric translation. In this method, the axes of the two datums are aligned using linear shifts of the X, Y and Z axes only. This transformation method is appropriate when the
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