Homological invariants and holorgraphic representations of topological structures in cellular spaces

Geometric modeling and computational representations of shapes have been subject to intense research for more than three decades. Interestingly, these subjects are still at the heart of a continuous activity of research and development in computer graphics, virtual environments, image-based rendering, computer-aided geometric design and physical simulations. Currently, geometric and physically-based modeling still face two main challenges: (1) the identification of topological features, and (2) the representation of the modes of interaction between them, both in static and dynamic environments. Current methods have offered many different forms of associating abstract structures with analytical expressions. The variety of modeling tools, from combinational methods to analytic algebraic geometry, not only reflects the richness of ideas in this domain of study but also the desire to improve, enhance and simplify. It is within this realm that we introduce a new framework, called holorgraphic geometric modeling (HGM). This framework combines the advantages of the graph-theoretic representation of combinatorial structures with the analytical flexibility, expressional power and scalability of higher-order, multi-dimensional variables and operators in the form of holors. HGM not only complements the combinatorial structures in geometric modeling but also enhances and reveals new concepts and ideas in the process of developing robust, flexible and scalable domains of formulation for simplicial complexes, cellular spaces, and homotopy in general.