By Herbert Edelsbrunner
This monograph provides a quick path in computational geometry and topology. within the first half the ebook covers Voronoi diagrams and Delaunay triangulations, then it provides the speculation of alpha complexes which play a vital position in biology. The important a part of the publication is the homology idea and their computation, together with the idea of patience that is imperative for functions, e.g. form reconstruction. the objective viewers contains researchers and practitioners in arithmetic, biology, neuroscience and desktop technology, however the booklet can also be helpful to graduate scholars of those fields.
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Additional info for A Short Course in Computational Geometry and Topology
Similar to the alpha complex filtration introduced in Chap. 5, we consider the growing sequence of shapes U(α), for α from 0 to ≤. 3 Pockets 43 Fig. 2 Eleven points arranged in the shape of the letter C. Left apart from the points, we see four unions of disks, with radii α0 < α1 < α2 < α3 . 3 Pockets This filtration enriches the shape description as we can now keep track of the holes as they appear and disappear during the thickening process. Indeed, the history of changes depends on the details of the original shape.
We now formally define the α-shape as the union of simplices in the α-complex. Note that the global connectivity of the union of disks in Fig. 4 is that same as that of the α-complex and of the α-shape: all three are connected and have a single hole. This is not a coincidence but rather a consequence of the Nerve Theorem, which will be discussed later in this course. 5 Filtration Next we vary α and consider the complete range of possible values, which is from 0 to ∞. For α < α ∀ , we have Ds (α) ∩ Ds (α ∀ ) and therefore Rs (α) ∩ Rs (α ∀ ).
Then we compute the Euler characteristic as the alternating sum of simplices: number of vertices minus number of edges plus number of triangles. The four triangulations shown in Fig. 3 all have 27 edges and 18 triangles, but they differ in the number of vertices, which from left to right is 9, 9, 10, 11. It follows that χ = 0, 0, 1, 2 for the torus, the Klein bottle, the projective plane, and the sphere. Remembering that the torus and the sphere are orientable, and the Klein bottle and the projective plane are not, we use these two 60 8 Topological Spaces Fig.
A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner