## 1 Introduction  
Coordinate transformation represents a change of coordinates from one coordinate reference system to another coordinate reference system based on a different datum through a one-to-one relationship. There are two important terms in this definition--coordinate system and datum. The coordinate system is a set of mathematical rules for specifying how coordinates are to be assigned to points, whereas datum represents a parameter or set of parameters that define the position of the origin, the scale, and the orientation of a coordinate reference system. The two most commonly used coordinate systems in geodesy are the Cartesian coordinate system, which gives the position of points relative to n multiple axis, and the ellipsoidal coordinate system, in which position is specified by geodetic latitude, geodetic longitude, and ellipsoidal height. Coordinate reference systems that use at least two independent coordinate reference systems are named compound coordinate reference systems [1]. With respect to scale along the coordinate axis, coordinate transformations can be divided in two main groups: affine and similarity transformations. Similarity transformations, which preserve angles, are characterized by unified scale along all coordinate system's axis, whereas this is not the case for affine transformation parameters, which do not preserve angles. Coordinate transformation procedures are also distinguished by the number of transformation parameters. In geodesy and geoinformatics, often used coordinate transformation procedures, due to the number of parameters, are: 3-parameter [2, 3], 5-parameter [2, 4], 7-parameter [2, 5, 6, 7], 8-parameter [8], 9-parameter [8], and 12-parameter [9]. Transformationparameters for all mentioned transformation procedures can be obtained from identical points in two different coordinate reference frames. The increase in the number of transformation parameters allows more errors linked to the realization of the coordinate system to be eliminated, but with a consequence of very complicated mathematical expressions coordinate transformation procedures. Many studies (e.g., [10; 11; 12; 13; 14]) considered different coordinate transformation procedures focusing on results but less on detail mathematical background, like derivation of numerical expressions. Therefore, this article focuses on a detail explanation of the aforementioned coordinate transformation procedures (mathematical derivations of transformation and inverse transformation equations along with numerical examples) for the example of the territory of the Republic of Croatia, utilizing coordinate transformation between the inherited coordinate reference system of the former Austrian-Hungarian Monarchy called Croatian State Coordinate System (cro. _Hrvatski drzavni koordinatui sustav_--HDKS) EPSG:3096 [15], with height component in Croatian Height Reference System 1875 (cro. _Hrvatski visinski referentni sustav 1875_--HVRS1875) EPSG:5195, and the official coordinate reference system--Croatian Terrestrial Reference System 1996 (cro. _Hrvatski terestricki referentni sustav 1996_--HTRS96) EPSG:4731 [16], with height component in Croatian Height Reference System 1971 (cro. _Hrvatski visinski referentni sustav 1971_--HVRS71) EPSG:5610, based on a set of 5194 identical points. Out of those identical 5194 points, 5000 were used for the calculation of the transformation parameters (Figure 1a), whereas the remaining 194 points were used for the external accuracy assessment. Points used for the external accuracy assessment were chosen in such a way that they are spread along the entire territory of the Republic of Croatia (Figure 1b). The height transformation between HVRS1785 and HVRS71 was excluded from this research in a way that all identical points in HDKS and HTRS96 have the same normal-orthometric height expressed in HVRS71. Transformation between HVRS1875 and HVRS71 is a part of the Croatian height transformation model [17], later included in the Croatian official HDKS/HTRS96 grid transformation model T7D [18].