The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through computational geometry, I stumbled upon a paper addressing the calculation of the 3D Winding Number for a closed triangular mesh. This method involves casting a ray in any direction and, for each triangle it intersects, computing the dot product between the ray's direction and the triangle's normal. If this value is positive, the winding number is decreased by 1; if negative, it is increased by 1. The paper posits that a point resides inside the mesh if the Winding Number equals 0; otherwise, it's outside if the number is nonzero.
This technique strikes me as somewhat similar to a Ray-Casting algorithm, another strategy for determining a point's position relative to a shape. Despite both methods aiming for the same outcome, I have been unable to find a theoretical link connecting them. Am I missing something, or is there a possible conceptual mix-up by the authors? Given that the paper underwent peer review, one would assume any such confusion would be flagged and corrected. Could anyone clarify if there is indeed a conceptual bridge between these two methods?
p.s. The definition of 3D Winding Number The article I am referring
I've resolved my query regarding the connection between the 3D Winding Number and Ray Casting methods. Traditionally, Ray Casting counts the number of times a ray crosses a polygon's edges to determine a point's inside/outside status by the parity of crossings. This approach aligns with the winding number for simply-connected polygons but diverges for self-intersecting polygons.
In 2001, Dan Sunday introduced a variation that mimics the winding number calculation without trigonometric or complex integrals. Like conventional Ray Casting, Sunday's method involves casting a ray from the point in question, typically in the +x direction, and tracking the polygon's tangent direction. However, it incorporates the crossing direction: incrementing the winding number for upward crossings and decrementing for downward ones. Completing a full trace adjusts the winding number based on these interactions, determining the point's status (inside if nonzero, outside if zero) with accuracy, even for self-intersecting polygons.
Applying this concept to 3D, by considering a cross-section plane that includes the point, effectively reduces the problem to a 2D scenario while preserving interior-exterior properties. The analogy in 3D involves using the dot product between the ray direction and the mesh normal, similar to assessing the crossing's direction. For consistent results, especially in mesh applications, ensuring that the normal vectors point outward and maintain consistent direction across neighboring triangles is crucial.
This understanding bridges the gap between theoretical winding number calculations and practical computational geometry applications, even in complex cases.