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Download Gauge Field Theory and Complex Geometry by Yuri Manin PDF

By Yuri Manin

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1 Approximation of Normals It turns out that the structure of the Voronoi cells contains information about normals. Indeed, if the sample is sufficiently dense, the Voronoi cells become long and thin along the direction of the normals at the sample points. 2). 1 (Medial). Let m 1 and m 2 be the centers of the two medial balls tangent to at p. The Voronoi cell V p contains m 1 and m 2 . 3. 2. Proof. Denote the medial ball with center m 1 as B. The ball B meets the surface only tangentially at points, one of which is p.

So, there is an alternate definition of local uniformity. A sample P is locally (ε, κ)-uniform for some ε > 0 and κ ≥ 1 if each point x ∈ has at least one and no more than κ points within ε f (x) distance. ˜ notation O(ε) Our analysis for different algorithms obviously involve the sampling parameter ε. To ease these analyses, sometimes we resort to O˜ notation which ˜ provides the asymptotic dependences on ε. A value is O(ε) if there exist two constants ε0 > 0 and c > 0 so that the value is less than cε for any positive ε ≤ ε0 .

A globally uniform sampling is more restrictive. It means that the sample is equally dense everywhere. Local feature size does not play a role in such sampling. There could be various definitions of globally uniform samples. We will say a sample 18 1 Basics P ⊂ is globally δ-uniform if any point x ∈ has a point in P within δ > 0 distance. In between globally uniform and nonuniform samplings, there is another one called the locally uniform sampling. This sampling respects feature sizes and is uniform only locally.

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