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# Trilateration Algorithm To Transform Cartesian Coordinates Into Geodetic Coordinates

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## Abstract

This research introduces and presents a new algorithm named “Trilateration Algorithm” to compute the Geodetic coordinates (ϕ, λ, h) of any point (P), given its Cartesian coordinates (xp, yp, zp) in reference to the geodetic ellipsoid. The algorithm is based on the new “Seta-Point Theorem” in the meridian plan, which defines a new deterministic Twin-Point (P0) for any point (P). From such a Twin-Point (P0), a single iteration solution is processed to achieve highly-accurate values for (ϕ, h) using a relatively simple and deterministic computational algorithm which is both valid and stable for all values of (ϕ, h). The proposed solution was tested on a sample of 4277 points. The mean error value of the computed latitude is -2.291E-10 arc-second and the mean error value of the computed height is 3.96E-8 mm.
INTRODUCTION
!
This! research! introduces! and! presents! a! new\$
algorithm\$ named! “Trilateration\$ Algorithm”! to! compute! the!
Geodetic!coordinates\$ (ϕ,\$ λ,\$ h)\$ of! any! point\$ (P),! given! its!
Cartesian\$ coordinates\$ (xp,\$ yp,\$ zp)! in! reference! to! the! geodetic!
ellipsoid.!!
The! algorithm!is!based\$ on\$ the! new! “Seta-Point\$
Theorem”! in! the! meridian\$ plan,! which! defines!a!new!
deterministic\$ Twin-Point\$ (P0)\$ for! any! point\$ (P).! From! such! a!
Twin-Point! (P0),! a! single\$ iteration\$ solution! is! processed! to!
achieve\$ highly-accurate\$ values\$ for! (ϕ,\$ h)\$ using! a! relatively!
simple\$ and\$ deterministic\$ computational\$ algorithm\$ which! is!
both!valid!and!stable!for!all\$values\$of\$(ϕ,\$h).!!
The! proposed\$ solution\$ was! tested! on! a! sample! of!
4277\$points.!The!mean\$error\$value!of!the!computed\$latitude\$is!
-2.291E-10\$ arc-second! and! the! mean\$ error\$ value! of! the!
computed\$height!is!3.96E-8\$mm.!
!
!
!References\$
\$
Featherstone,!W.,!&!Claessens,!S.!(2008).!Closed-form!transformation!between!geodetic!and!ellipsoidal!coordinates.!Studia!Geophysica!et!Geodaetica,!52(1),!1–18.!
Ligas,!M.,!&!Banasik,!P.!(2011).!Conversion!between!Cartesian!and!geodetic!coordinates!on!a!rotational!ellipsoid!by!solving!a!system!of!nonlinear!equations.!Geodesy!
and!Cartography,!60(2),!145-159.!
Heiskanen,!W.,!&!Moritz,!H.!(1967).!Physical!Geodesy.!San!Francisco:!Freeman!and!Co.!
Trilateration Algorithm To Transform Cartesian Coordinates Into Geodetic Coordinates
Mohamed ELEICHE(1,*) and Ahmed Hamdi MANSI(2,3)
(1) Faculty of Engineering, Egyptian Russian University, Cairo, Egypt; (*) Corresponding Author: mohamed.eleiche@gmail.com;
(2) Civil Engineering Department, College of Engineering, Shaqra University, Dawadmi, Saudi Arabia (ahmed.mansi@su.edu.sa);
(3) Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Pisa, Italy (ahmed.mansi@INGV.it).
PROCESSING ALGORITHM
1.\$Given\$(r,\$z)\$and!required\$(ϕ,\$h):!
any!point\$(P)\$has!an!angle\$(ϕ)!and!
a!height\$(h)\$above!the!reference!
ellipsoid;!!
2.Compute\$the\$Seta\$Point\$(W):!the!
Line\$PW\$crosses!the!ellipse!in!
point\$(Qw);!
3.The! Line\$ PW\$ makes! an! angle\$ (ϕw)\$ with! a! horizontal\$
distance\$(hw)\$from!point\$(P)\$to!the!intersection\$(Qw);!
!
4.The! deterministic\$ Twin-Point\$ (P0),\$
defined!by!w,\$hw),!is!very!close!to!the!
point\$(P).!
RESULTS and DISCUSSION
!
CONCLUSION
!
The! Seta-Point\$ theorem\$ provides! a! new\$ trilateration\$
algorithm\$ for! the! conversion!of!Cartesian!coordinates! into!
Geodetic\$coordinates.!!
The! computational\$ requirements\$ of! the! new! algorithm! are!
simple! and! do! not! exceed! the! solution\$ of\$ three\$ quadratic\$
functions\$for!the!points!located!on!the!ellipse!surface.!!
The! algorithm!is!valid! for! all! values\$ of\$ geodetic\$ and!
Cartesian\$ coordinates.! Two\$ iterations\$ would! generate!
highly\$accurate\$geodetic\$coordinates.!
1.Each!iteration!is!closer!to!the!true\$value;!
2.The!relation!between!latitude!(ϕ)! versus!computational!
error!of!the!latitude\$(ϕ-ϕw)\$and!height\$(h-hw).!
3.The!Boxplot!for!the!latitude\$error!(ϕ-ϕw).!The!Seta-point!
is!at!the!left,!the!first!iteration!in!the!middle,!and!the!
third!iteration!at!right;!
4.The!Boxplot!for!the!height\$error!(h-hw).!The!Seta-point!is!
at!the!left,!the!first!iteration!in!the!middle,!and!the!third!
iteration!at!right;!
Pollard,!J.!(2002).!Iterative!vector!methods!for!computing!geodetic!latitude!from!rectangular!coordinates.!Journal!of!Geodesy,!60,!36-40.!
Shu,!C.,!&!Li,!F.!(2010).!An!iterative!algorithm!to!compute!geodetic!coordinates.!Computers!&!Geosciences,!1145-1149.!
Vermeille,!H.!(2002).!Direct!transformation!from!geocentric!coordinates!to!geodetic!coordinates.!Journal!of!Geodesy,!76,!451-454.!
ResearchGate has not been able to resolve any citations for this publication.
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By using the Newton–Raphson method to solve a quartic equation of the Lagrange parameter, we propose a new iterative algorithm for the transformation from Cartesian to geodetic coordinates. Numerical experiments show that the new method is sufficiently precise and free from singularity and non-convergency, except within 50 km of the geocentre. After one iteration, the maximum error of latitude is less than 10−8 arc-sec and the relative height error is less than 10−15 over the range of geodetic heights from −106 to 1012 m. Comparisons of the complexity and performance with some other well-known methods indicate that the algorithm is efficient and accurate.