Application of Polynomial Transformation in Georeference of Graphic Cadastral Plans

Polynomial transformation is a classic mathematical tool in geodesy, used to convert coordinates between different reference systems. It is mainly used in cases where the exact geometric or physical model of the relationship between the systems is unknown, as well as in the presence of significant local deformations in the source data. In practice, this method is used in: transformation between coordinate systems (including local), approximation of heights when creating geoid models, localization in GNSS measurements, as well as in specific tasks in photogrammetry and laser scanning. One of its most widespread applications remains the georeferencing of raster images. This paper investigates the specifics of applying polynomial transformation for georeferencing purposes. The main emphasis is placed on the dependence of the results on the configuration and the number of control points used. An analysis is made and recommendations are formulated for their optimal distribution on the image. The mathematical requirement for uniform grid coverage in a minimum of n+1 columns and rows (where n is the degree of the polynomial) is justified as a condition for avoiding singularity. Additionally, the negative effects of the collinearity of the control points on the computational stability are considered, as well as the approaches for identifying the Runge phenomenon (appearance of high-amplitude oscillations) in the extrapolation zones along the boundaries of the transformed range.