Polar Fitting and Hermite Interpolation for Freeform Droplet Geometry Measurement

Droplet-based microlens fabrication using UV-curable polymers demands precise meas-urement of three-dimensional geometries, especially for nonaxisymmetric shapes influ-enced by electric field deformation. In this work, we present a polar coordinate-based con-tour fitting method combined with Hermite interpolation to reconstruct 3D droplet geom-etries from two orthogonal shadowgraphy images. The image segmentation process inte-grates superpixel clustering with active contours to extract the droplet boundary, which is then approximated using a spline-based polar fitting approach. The two resulting con-tours are merged using a polar Hermite interpolation algorithm, enabling the reconstruc-tion of freeform droplet shapes. We validate the method against both synthetic CAD data and precision-machined reference objects, achieving volume deviations below 1% for ax-isymmetric shapes and approximately 3.5% for nonaxisymmetric cases. The influence of focus, calibration, and alignment errors is quantitatively assessed through Monte Carlo simulations and empirical tests. Finally, the method is applied to real electrically de-formed droplets, with volume deviations remaining within the experimental uncertainty range. This demonstrates the method’s robustness and suitability for metrology tasks in-volving complex droplet geometries.