New PDF release: Computational Topology - An Introduction

By Herbert Edelsbrunner

Combining ideas from topology and algorithms, this ebook offers what its identify provides: an creation to the sector of computational topology. beginning with motivating difficulties in either arithmetic and computing device technological know-how and build up from vintage subject matters in geometric and algebraic topology, the 3rd a part of the textual content advances to power homology. This viewpoint is significantly vital in turning a normally theoretical box of arithmetic into one who is correct to a large number of disciplines within the sciences and engineering. the most procedure is the invention of topology via algorithms. The publication is perfect for instructing a graduate or complicated undergraduate path in computational topology, because it develops the entire history of either the mathematical and algorithmic points of the topic from first ideas. therefore the textual content may perhaps serve both good in a path taught in a arithmetic division or desktop technological know-how division.

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More generally, we have a 4g-gon for a sphere with g tubes and a 2g-gon for a sphere with g cross-caps attached to it. 3. Note that the square of the torus is in standard form but that of the Klein bottle is not. Classification Theorem for Compact 2-manifolds. The two infinite families S2 , T2 , T2 #T2 , . . and P2 , P2 #P2 , . . exhaust the family of compact 2-manifolds without boundary. The first family of orientable, compact 2-manifolds consists of the sphere, the torus, the double torus, and so on.

Let G = (V, E) be a simple, undirected graph. A drawing maps every vertex u ∈ V to a point f (u) in R2 , and it maps every edge uv ∈ E to a path with endpoints f (u) and f (v). The drawing is an embedding if the points are distinct, the paths are simple and do not cross each other, and incidences are limited to endpoints. Not every graph can be drawn without crossings. The graph is planar if it has an embedding in the plane. 12 for the complete graph of four vertices, there are many drawings of a planar graph, some with and some without crossings.

We usually envision them put into three-dimensional space, sometimes with and preferably without selfintersections. Not all surfaces can be embedded in three-dimensional Euclidean space and self-intersections are unavoidable, but often they are accidental. Indeed, choosing a nice embedding of a surface in space is an interesting computational problem. We address this question for surfaces made out of triangles. 1 II Surfaces Two-dimensional Manifolds In our physical world, the use of the term surface usually implies a 3dimensional, solid shape of which this surface is the boundary.

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Computational Topology - An Introduction by Herbert Edelsbrunner


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