Modelado de trayectorias de drones mediante interpolación matemática para el análisis de la precisión de medición de superficies planas

The use of drones in topographic surveys has increased efficiency and coverage in surface measurements; however, the accuracy of the results depends critically on flight altitude, trajectory type, and the mathematical processing of spatial data. In this context, the objective of this study was to model drone trajectories using Lagrange polynomial interpolation in order to analyze their influence on the accuracy of flat-surface measurements. The methodology adopted a quantitative experimental–analytical approach, evaluating area measurement errors obtained through drone-based photogrammetry at different flight altitudes (50 m, 80 m, and 120 m) and comparing them with a traditional reference method. Nonparametric statistical tests were applied to identify significant differences, and geometric models of linear and parabolic trajectories were developed using regression techniques, least squares, and Lagrange interpolation, incorporating simulations with different node configurations and error metrics such as RMSE and MAE. The results showed that a flight altitude of 80 m minimizes measurement error and that interpolated parabolic trajectories exhibit significantly superior performance compared to rectilinear paths. Furthermore, low-degree Lagrange interpolation using three representative nodes was shown to provide the best balance between accuracy and numerical stability, avoiding the Runge phenomenon associated with high-order polynomials. In conclusion, the study confirms that appropriate mathematical modeling of drone trajectories substantially improves topographic accuracy and provides objective criteria for the planning of photogrammetric flights.