Disjoint convex shell and its applications in mesh unfolding. (September 2017) - Kim, Yun-hyeong; Xi, Zhonghua; Lien, Jyh-Ming
Abstract: In this work, we study a geometric structure called disjoint convex shell or simply DC-shell. A DC-shell of a polyhedron is a set of pairwise interior disjoint convex objects that collectively approximate the given polyhedron. Preventing convex objects from overlapping enables faster and robust collision response and more realistic fracturing simulation. Without the disjointness constraint, a physical realization of the approximation becomes impossible. This paper investigates multiple approaches that construct DC-shells from shapes that are either composed of overlapping components or segmented into parts. We show theoretically that, even under this rather simplified setting, constructing DC-shell is difficult. To demonstrate the power of DC-shell, we studied how DC-shell can be used in mesh unfolding, an important computational method in manufacturing 3D shape from the 2D material. Approximating a given polyhedron model by DC-shells provides two major benefits. First, they are much easier to unfold using the existing unfolding methods. Second, they can be folded easily by both human folder or self-folding machines. Consequently, DC-shell makes paper craft creation and design more accessible to younger children and provides chances to enrich their education experiences. Graphical abstract: Highlights: DC-shell is a set of disjoint convex objects approximating non-convex overlapping sets. DC-shell provides better approximation than those created by other methods. DC-shell enables faster collision response and more realistic fracturing simulation than overlapping convex objects do. DC-shell can be easily and rigidly unfolded into high-quality nets.