Theoretical foundations of point cloud coordinate system transformation

Abstract

Purpose. To provide theoretical foundations and develop mathematical models for the efficient transformation of coordinate systems for point clouds in geophysical research; the scientific analysis is aimed at developing algorithms and establishing necessary dependencies for the reliable integration of data obtained at different time points into a unified coordinate system, opening up prospects for further study and analysis of processes in geophysical research. The methods.The calculation is carried out using the following steps. Determination of known coordinates of four points (x1', y1', z1'; x2', y2', z2'; x3', y3', z3'; x4', y4', z4') in a hypothetical coordinate system (X', Y', Z') and the coordinates of the same points (x1, y1, z1; x2, y2, z2; x3, y3, z3; x4, y4, z4) in the coordinate system (X, Y, Z) to which the point clouds need to be transformed. Determination of constants a1, a2, a3, d, b1, b2, b3, e, c1, c2, c3, f through a system of equations. After determining the constants, the coordinates of points (x', y', z') in the hypothetical coordinate system (X', Y', Z') are calculated using equations where each equation expresses the coordinates of points (x', y', z') in terms of coordinates of points (x, y, z) in the coordinate system (X, Y, Z) and the determined constants. After performing the calculations, point clouds can be merged into a single coordinate system using the computed coordinates (x', y', z'). This methodology allows for the successful transformation of coordinate systems for point clouds in geophysical research. Findings. Analytical regularities have been established based on known coordinates of four points in both coordinate systems, allowing for the efficient transformation of a point cloud from one coordinate system to another. The originality. For the first time, precise analytical dependencies have been established that enable the efficient transformation of point clouds from one coordinate system to another using known coordinates of four points in both systems. Practical implementation. The obtained dependencies enable the efficient transformation of point clouds from one coordinate system to another using known coordinates of four points in both systems.