On a Fundamental Physical Principle Underlying the Point Location Algorithm in Computer Graphics

The issue of point location is an important problem in computer graphics and the study of eecient data structures and fast algorithms is an important research area for both computer graphics and computational geometry disciplines. When lling the interior region of a planar polygon in computer graphics, it is necessary to identify all points that lie within the interior region and those that are outside. Sutherland and Hodgman are credited for designing the rst algorithm to solve the problem. Their approach utilizes vector construction and vector cross products, and forms the basis of the \odd parity" rule. To verify whether a test point is within or outside a given planar polygon , a ray from the test point is drawn extending to innnity in any direction without intersecting a vertex. If the ray intersects the polygon outline an odd number of times, the region is considered interior. Otherwise, the point is outside the region. In 3 dimensional space, Lee and Preparata propose an algorithm but their approach is limited to point location relative to convex polyhedrons with vertices in 3D-space. Although it is rich on optimal data structures to reduce the storage requirement and eecient algorithms for fast execution, a proof of cor-rectness of the algorithm, applied to the general problem of point location relative to any arbitrary surface in 3D-space, is absent in the literature. This paper argues that the electromagnetic eld theory and Gauss's Law constitute a fundamental basis for the \odd parity" rule and shows that the \odd parity" rule may be correctly extended to point location relative to any arbitrary closed surface in 3D-space.