Decompositions of polyhedra in three dimensions

This thesis deals with new theoretical and practical results on convex and CSG decompositions, and triangulations of polyhedra in three dimensions. Convex and CSG decompositions of polyhedra find applications in simpler algorithms in motion planning, computer graphics, and solid modeling. Triangulations of polyhedra are fundamental nontrivial steps in finite element simulations and CAD/CAM applications. To reduce ill conditioning as well as discretization error in finite element simulations, near regular shaped elements are desired. This motivates triangulation algorithms for polyhedra that produce well shaped tetrahedra. We present efficient algorithms for convex and CSG decompositions of polyhedra with arbitrary genus. A modification of this decomposition method gives an efficient algorithm for triangulations of polyhedra. The efficiency of these algorithms is mainly derived from the use of zone theorem on hyperplane arrangements, studied in combinatorial geometry. A triangulation algorithm that triangulates a convex polyhedron and a three dimensional point set, in general, with guaranteed quality tetrahedra is also presented. In particular, this algorithm guarantees that four out of five possible bad tetrahedra are never generated. Geometric algorithms, when implemented under finite precision arithmetic often crash or fail to produce valid output because of numerical errors. We have investigated this problem of output inconsistency under imprecise arithmetic computations in order to provide topologically robust implementations of the decomposition algorithms. Implementations are carried out as part of SHILP, a solid modeling and display toolkit that runs on Unix workstations under the X Window System

Triangulating with high connectivity

We consider the problem of triangulating a given point set, using straight-line edges, so that the resulting graph is "highly connected." Since the resulting graph is planar, it can be at most 5-connected. Under the nondegeneracy assumption that no three points are collinear, we characterize the point sets with three vertices on the convex hull that admit 4-connected triangulations. More generally, we characterize the planar point sets that admit triangulations having neither chords nor noncomplex (i.e., nonfacial) triangles. We also show that any planar point set can be augmented with at most 2 extra points to admit a 4-connected triangulation. All our proofs are constructive, and the resulting triangulations can be constructed in 0(n log n) time. We conclude by stating several open problems

Fabrication of a Vitamin B12-Loaded Carbon Dot/Mixed-Ligand Metal Organic Framework Encapsulated within the Gelatin Microsphere for pH Sensing and In Vitro Wound Healing Assessment

Bacterial invasion is a serious concern during the wound healing process. The colonization of bacteria is mainly responsible for the pH fluctuation at the wound site. Therefore, the fabrication of a proper wound dressing material with antibacterial activity and pH monitoring ability is necessary to acquire a fast healing process. Therefore, this work is dedicated to designing a vitamin B12-loaded gelatin microsphere (MS) decorated with a carbon dot (CD) metal–organic framework (MOF) for simultaneous pH sensing and advanced wound closure application. The resultant MS portrayed a high specific surface area and a hierarchically porous structure. Furthermore, the surface of the resultant MS contained numerous carboxyl groups and amine groups whose deprotonation and protonation with the pH alternation are accountable for the pH-sensitive properties. The vitamin B12 release study was speedy from the MOF structure in an acidic medium, which was checked by gelatin coating, and a controlled drug release behavior was observed. The system showed excellent cytocompatibility toward the L929 cell line and remarkable antibacterial performance against Gram-negative Escherichia coli and Gram-positive Staphylococcus aureus. Furthermore, the combined effect of Zn2+, the imidazole unit, and CDs produces an outstanding bactericidal effect on the injury sites. Finally, the in vitro wound model suggests that the presence of the vitamin B12-loaded gelatin MS accelerates the proliferation of resident fibroblast L929 cells and causes tissue regeneration in a time-dependent manner. The relative wound area, % of wound closure, and wound healing speed values are remarkable and suggest the requirement for assessing the response of the system before exploiting its prospective in vivo application