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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!(ϕw,$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.!