Equivalence properties of 3D conformal transformations and their application to reverse transformations

Abstract

Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can be constructed in more than one way. Two possible orderings of the rotations are in common use, giving rise to Helmert versions 1 and 2. It is demonstrated how the rotation parameters of either version can be converted into the rotation parameters of the other. This is useful when software is designed for the other version. It also enables computation of the same-formula inverse transformation by changing the sign of the equivalent 'other version' parameters. These results were primarily intended for conformal transformations between geodetic datums. They can, however, be extended to coordinate transformations in disciplines such as photogrammetry where rotations sometimes exceed 90 degrees.

Method of Improving Incomplete Spatial-Temporal Data in Inland Navigation, on the Basis of Industrial Camera Images – West Oder River Case Study

“…Coordinate transformations in orthogonal reference systems are generally based on transformations by similarity, called also Helmert or conformal transformation (Baetslé, 1966;Späth, 2004;Deakin, 2006;Ruffhead, 2020). This transformation is three-dimensional, seven-parameters transformation and can be presented in the form of a relation (Deakin, 1998…”

Detail Explanation of Coordinate Transformation Procedures Used in Geodesy and Geoinformatics on the Example of the Territory of the Republic of Croatia

“…Conducting the standard and abridged inverse transformation is not possible in a direct way because inverse transformation depends on the values φ i and λ i , which are unknown in the output coordinate system. In order to perform the inverse transformation, Ruffhead [42] gives the next solution:…”

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“…Ker pa so zasuki med globalnimi referenčnimi sestavi zelo majhni, lahko enačbo lineariziramo. Linearizirana enačba je enaka za obe različici stroge enačbe (Ruffhead, 2021). Se pa za linearizirano različico enačbe pojavljata dve različni interpretaciji parametrov rotacijske matrike.…”

“…Za parametre rotacijske matrike je uporabljena PV-konvencija (angl. position vector transformation), ki jo je skladno s priporočilom mednarodnega standarda Geografske informacije -lociranje s koordinatami (ISO 19111, 2019) sprejela tudi Mednarodna zveza za geodezijo (Ruffhead, 2021). Za navedeno enačbo transformacije velja, da lahko za poenostavljeno (približno) obratno transformacijo uporabimo kar isto enačbo in parametre z nasprotnimi predznaki (angl.…”

“…Argument za uporabo strogih enačb pri obratni transformaciji je izogibanje kumulativnim napakam pri večkratnih transformacijah iz izvornega v ciljni sestav in nazaj -ko na primer pri transformaciji iz ITRF2014 v D17 vektor hitrosti določamo iz geokinematskega modela in moramo uporabiti iterativni postopek (podpoglavje 3.2, postopek B). Nezanemarljiv je tudi učinek povratnosti transformacije na zaupanje uporabnikov v programsko opremo (Ruffhead, 2021).…”