By Herbert Edelsbrunner
This monograph offers a brief path in computational geometry and topology. within the first half the e-book covers Voronoi diagrams and Delaunay triangulations, then it provides the speculation of alpha complexes which play a vital position in biology. The vital a part of the e-book is the homology conception and their computation, together with the speculation of patience that is vital for functions, e.g. form reconstruction. the objective viewers includes researchers and practitioners in arithmetic, biology, neuroscience and machine technological know-how, however the publication can also be necessary to graduate scholars of those fields.
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Extra resources for A Short Course in Computational Geometry and Topology
5, we see tunnels going in parallel through the entire structure. On the right, we see the pockets of the same structure. Many of them have two mouths and can thus be identified as geometric representations of tunnels. In later sections, we will use the ranks of homology groups to count tunnels or, more precisely, to measure the size of a basis that generates all possible ways to pass through the shape following tunnels. References 1. Edelsbrunner H, Facello MA, Liang J (1998) On the definition and the construction of pockets in macromolecules.
The diagram is obtained by taking the union of the balls centered at the atoms in which the radii are chosen so that the atoms are at equilibrium when the balls touch. Different types of atoms affect neighboring atoms differently, which leads to different radii. For example, hydrogen atoms are the smallest, with carbon, oxygen, and nitrogen atoms represented by somewhat larger balls. This motivates the concept of weighted alpha complexes, which are defined analogous to weighted Voronoi diagrams and weighted Delaunay triangulations.
4 Voronoi Decomposition To get a cleaner relationship between the union of disks and the α-shape, we need an unambiguous definition of the latter. For this, we overlay the union of disks with the Voronoi diagram, effectively decomposing the union into convex regions; see Fig. 3. To formalize this idea, we write Rs (α) = Vs ◦ Ds (α) and note that this is a convex set because it is the intersection of convex sets. Furthermore, U S (α) = s√S Rs (α). In words, the regions Rs cover the union, but in contrast to the disks, which also cover the union, they do this without overlap.
A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner