Addressing the Issues in the Winding Number Algorithm Related to the Floating Point Representation

The winding number has proven to be a very simple and robust approach for inside-outside segmentation that can be applied both in the 2D and 3D realms for multiple purposes, including mesh cleaning, CAD design, 3D printing, among others. However, the error introduced by the floating point representation can lead to misclassification of points, what can in turn lead to further errors in the computation. The potential scenarios for which the winding number could lead to misclassification are identified in this research both for 2D and 3D, and these faulty scenarios are explored by performing a series of experiments. These scenarios can be very particular or rare, but in which misclassification can lead to the wrong segmentation of a space subdivision. In addition, a solution to address these issues is proposed by implementing a more robust version of the 2D winding number using rational arithmetic and a slightly more robust version of the 3D winding number by using multi-precision representation instead of floating point. This research also introduces a novel approach for inside-outside segmentation based on least squares that not only addresses the floating point misclassification issue in 3D, but also does it at a fast speed and with exceptional results, making it a great choice for mesh reconstruction and for extracting closed watertight meshes.