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This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this… Expand
Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that… Expand
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram… Expand
We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as… Expand
Identification and size characterization of surface pockets and occluded cavities are initial steps in protein structure-based ligand design. A new program, CAST, for automatically locating and… Expand
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical… Expand
A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to… Expand
A set ofn weighted points in general position in ℝ d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will… Expand
AbstractWe formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during… Expand
1. Delaunay triangulations 2. Triangle meshes 3. Combinatorial topology 4. Surface simplification 5. Delaunay tetrahedrizations 6. Tetrahedron meshes 7. Open problems.