Numerical Shape Interrogation of Planar Generalized Cubic Curves. (May 2022) - Gabrielides, Nikolaos C.; Sapidis, Nickolas S.
Abstract: The authors in Gabrielides and Sapidis (2020) presented a shape analysis of planar as well as of spatial Generalized Cubic Curves (GCC). The present paper, focusing on planar GCCs, complements (Gabrielides and Sapidis, 2020) with vital new results, offering necessary and sufficient conditions to directly identify the cases where a GCC is convex, non-convex or irregular, and/or if it presents a loop. These conditions lead to a simple and efficient algorithm, fully detailed in this paper, performing comprehensive shape interrogation of a GCC. Graphical abstract: Highlights: Generalized cubic curves: representation with totally positive bases. Rate of convergence of control polygon to a generalized cubic curve under subdivision. Planar generalized cubic curves with hyper-convex control polygons. Detection of inflections, cusps, and loops in planar generalized cubic curves.