Computational Topology for Geometric Design and Molecular Design

The nascent field of computational topology holds great promise for resolving several long-standing industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, aerospace, shipbuilding and consumer product industries. Commercial CAGD packages depend upon complementary geometric and topological algorithms. The emergence of geometric modeling for molecular simulation and pharmaceutical design presents new challenges for supportive topological software within Computer Aided Molecular Design (CAMD) systems. For both CAGD and CAMD systems, splines provide relatively mature geometric technology. However, there remain pernicious issues regarding the ‘topology’ of these models, particularly for support of robust simulations which rely upon the topological characteristics of adjacency, connectivity and non-self-intersection. This paper presents current challenges and frontiers for reliable simulation and approximation of topology for geometric models. The simultaneous consideration of CAGD and CAMD is important to provide unifying abstractions to benefit both domains. In engineering applications it is a common requirement that topological equivalence be preserved during geometric modifications, but in molecular simulations attention is focused upon where topological changes have occurred as indications of important chemical changes. The methods presented here are supportive of both these disciplinary approaches. University of Connecticut, Department of Computer Science and Engineering University of Connecticut, Department of Computer Science and Engineering, Department of Mathematics. Partial funding for both authors was from National Science Foundation grants DMS0138098, CCR 0226504 and CCF 0429477. All statements in this publication are the responsibility of the authors, not of the National Science Foundation. “mi03 ̇5 ̇10” 2005/3/20 page i i