Download A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner PDF

By Herbert Edelsbrunner

This monograph offers a brief direction in computational geometry and topology. within the first half the publication covers Voronoi diagrams and Delaunay triangulations, then it provides the idea of alpha complexes which play an important position in biology. The valuable a part of the ebook is the homology conception and their computation, together with the speculation of patience that is imperative for functions, e.g. form reconstruction. the objective viewers contains researchers and practitioners in arithmetic, biology, neuroscience and computing device technological know-how, however the booklet can also be helpful to graduate scholars of those fields.

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Extra resources for A Short Course in Computational Geometry and Topology

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4) We call such an increasing sequence as a filtration. The not necessarily contiguous subsequence of alpha complexes is sometimes referred to as the alpha complex 38 5 Alpha Complexes filtration of the set S. This and other filtrations will play an important role in later sections, when we talk about persistent homology. 6 Space-Filling Models of Proteins A major application of alpha complexes are proteins and other molecules modeled as unions of balls [4]. One such model is the van der Waals diagram of a protein.

The torus and the sphere can be embedded in R3 , so we are quite familiar with their curved appearance, but the Klein bottle and the projective plane cannot, which is perhaps the reason why they are more difficult to imagine. The projective plane seems most difficult to imagine of all, so we offer an alternative construction. Starting with the sphere, S2 = {x ≥ R3 | √x√ = 1}, we call points x and −x antipodal. Gluing the antipodal points in pairs gives the projective plane. This recipe glues the northern hemisphere to the southern hemisphere, and it glues the equator to itself, like wrapping a hair-band twice around a pony-tail.

This part focuses on the homology of individual topological spaces, which is a classic topic in algebraic topology. Chapter 8 Topological Spaces In geometry, we can decide which points are near and which are far by computing distance. In topology, no such notion is available and distance is replaced by the weaker concept of neighborhoods. A full treatment of this idea is beyond the scope of this course. Indeed, we are motivated to side-step the technical difficulties by restricting ourselves to spaces in which connectivity can be defined by elementary means.

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