Article

Trilateration algorithm to transform Cartesian coordinates into geodetic coordinates

Authors:
To read the full-text of this research, you can request a copy directly from the author.

Abstract

The transformation of Cartesian coordinates (xp, yp, zp) of a point (P) into their geodetic equivalent (φ, λ, h) in reference to the geodetic ellipsoid, is an essential requirement in geodesy. There are many well-known algorithms solving this transformation in closed-form, approximate or iterative approaches. This paper presents a new algorithm named “Trilateration Algorithm” for this transformation. It is based on the new “Seta-Point Theorem” in the meridian plan, which defines a new deterministic Twin-Point (P0) for the point (P). From the Twin-Point (P0), a single iteration solution is processed to achieve highly-accurate values for (φ, h) in a relatively simple and deterministic computation algorithm which is valid and stable for all values of (φ, h). The proposed solution was tested on a sample of 4277 points that cover all possible cases of point (P). The produced maximum absolute error in latitude is (0.000 0026″) and (0.000 476 mm) in height, computed from first iteration.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the author.

... The Fukushima-Halley algorithm is the algorithm created by Fukushima accelerated by Halley method (Fukushima 2006) and is considered the standard in the geodetic conversion and was proven to be the fastest among other well known algorithms (Ligas and Banasik 2011). The Trilateration algorithm (Eleiche 2020) is among the latest one in this series of algorithms. It is important to measure the accuracy and efficiency of any new algorithm (Claessens 2019) through extensive test set of points (Panou 2019). ...
... We developed a Visual C++ program for both algorithms and applied it on the same dataset of 4277 points to measure the precision of both algorithms. This dataset was previously implemented in the validation of the Trilateration algorithm (Eleiche 2020). For each point (P) with elliptical coordinates (φ, h), the program computed the equivalent Cartesian coordinates (X,Y,Z), which will be the input for the transformation program to compute (φ c , h c ). ...
Article
Full-text available
The geodetic transformation of Cartesian coordinates into their elliptical equivalent is a fundamental problem in geodesy. The Fukushima algorithm accelerated by Halley method (Fukushima-Halley) is considered the standard in this conversion. The Trilateration algorithm is a recent algorithm solving the conversion problem through a computational geometry approach. This study compared the Trilateration algorithm to the Fukushima-Halley algorithm in aspects of accuracy of results, time efficiency, and space efficiency. Also, the parallel version of both algorithms was established using the Master-Slave technique and compared. The Trilateration Algorithm showed a slightly higher accuracy compared to Fukushima-Halley algorithm, which allocated less space in memory, and was 2.6 faster in sequential version compared to 1.9 in the parallel version. The study introduced a benchmark for arithmetic operation on the testing machine to be used in time efficiency comparison.
... By using the linear equation and the expression of the perpendicular distance from the origin, the point (X i , Y i ) in the image space is mapped to the accumulator HT(σ i , ϖ i ) in the Hough space, and all points in the image space where the two formulas are true are numerically added to the corresponding accumulator to realize the calculation of the Hough transform algorithm [13,14]. If there is a line in the image, then the accumulator has a local maximum. ...
Article
Full-text available
Current image recognition methods cannot combine the transmission of image data with the interaction of image features, so the steps of image recognition are too independent, and the traditional methods take longer time and cannot complete the image denoising. Therefore, a recognition method of sports training action image based on software defined network (SDN) architecture is proposed. The SDN architecture is used to integrate the image data transmission and interactive process and to optimize the image processing centralization. The network architecture is composed of application layer, control layer, and infrastructure layer. Based on this, the dimension of image sample set is reduced, and the edge detection operator in any direction is constructed. The image edge filter is realized by calculating the response and threshold of image edge by using lag threshold and nonmaximum suppression (NMS). The Hough transform algorithm is improved to optimize the detection range. Extracting the neighborhood feature of sports training action, the recognition of sports training action image based on SDN architecture is completed. Simulation results show that the proposed method takes less time and the image denoising effect is better. In addition, the F1 test results of the proposed method are higher than those of the literature, and the convergence is better. Therefore, the performance of the proposed method is better.
Article
Full-text available
Ghana a developing country still adopt the non-geocentric ellipsoid known as the War Office 1926 as its horizontal datum for all surveying and mapping activities. Currently, the Survey and Mapping Division of Lands Commission in Ghana has adopted the satellite positioning technology such as Global Positioning System based on a geocentric ellipsoid (World Geodetic System 1984 (WGS84)) for its geodetic surveys. It is therefore necessary to establish a functional relationship between these two different reference frames. To accomplish this task, the Bursa-Wolf transformation model was applied in this study to obtain seven transformation parameters namely; three translations, three rotations and a scale factor. These parameters were then used to transform the WGS84 data into the War office system. However, Ghana’s national coordinate system is a projected grid coordinate and thus the new War Office coordinates (X, Y, Z) obtained are not applicable. There is therefore the need to project these coordinates onto the transverse Mercator of Ghana. To do this, the new war office data (X, Y, Z) attained must first be transformed into geodetic coordinates. The reverse conversion from cartesian (X, Y, Z) to its corresponding geodetic coordinate (φ, λ, h) is computation intensive with respect to the estimation of geodetic latitude and height. This study aimed at evaluating the performance of seven methods in transforming from cartesian coordinates to geodetic coordinates within the Ghana Geodetic Reference Network. The seven reverse techniques considered are Simple Iteration, Bowring Inverse equation, method of successive substitution, Paul’s method, Lin and Wang, Newton Raphson and Borkowski’s method. The obtained results were then projected onto the transverse Mercator projection to get the new projected grid coordinates in the Ghana national coordinate system. These results were compared with the existing coordinates to assess their performance. The authors proposed the Paul’s method to be a better fit for the Ghana geodetic reference network based on statistical indicators used to evaluate the reverse methods performance.
Article
Full-text available
A new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 - 10 km) (terrestrial), (20 - 1000 km), and (1000 - 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σφ < 0.0004") and height (σ h < 10-6 km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.
Article
Full-text available
A singularity-free perturbation solution is presented for inverting the Cartesian to Geodetic transformation. Geocentric latitude is used to model the satellite ground track position vector. A natural geometric perturbation variable is identified as the ratio of the major and minor Earth ellipse radii minus one. A rapidly converging perturbation solution is developed by expanding the satellite height above the Earth and the geocentric latitude as a perturbation power series in the geometric perturbation variable. The solution avoids the classical problem encountered of having to deal with highly nonlinear solutions for quartic equations. Simulation results are presented that compare the solution accuracy and algorithm performance for applications spanning the LEO-to-GEO range of missions.
Article
Full-text available
A new iterative procedure to transform geocentric rectangular coordinates to geodetic coordinates is derived. The procedure solves a modification of Borkowski's quartic equation by the Newton method from a set of stable starters. The new method runs a little faster than the single application of Bowring's formula, which has been known as the most efficient procedure. The new method is sufficiently precise because the resulting relative error is less than 10−15, and this method is stable in the sense that the iteration converges for all coordinates including the near-geocenter region where Bowring's iterative method diverges and the near-polar axis region where Borkowski's non-iterative method suffers a loss of precision.
Article
We present formulas for direct closed-form transformation between geodetic coordinates (φ, λ, h) and ellipsoidal coordinates (β, λ, u) for any oblate ellipsoid of revolution. These will be useful for those dealing with ellipsoidal representations of the Earth’s gravity field or other oblate ellipsoidal figures. The numerical stability of the transformations for near-polar and near-equatorial regions is also considered.
Article
 The transformation from geocentric coordinates to geodetic coordinates is usually carried out by iteration. A closed-form algebraic method is proposed, valid at any point on the globe and in space, including the poles, regardless of the value of the ellipsoid's eccentricity.
Article
An exact and relatively simple analytical transform of the rectangular coordinates to the geodetic coordinates is presented. It does not involve any approximation and the accuracy of practical calculations depends exclusively on the round-off errors. The algorithm is based on one solution to the quartic equation in tg(45-/2), where is the parametric (or eccentric) latitude.