Journal ArticleDOI
Inequalities for the Curvature of Curves and Surfaces
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The difference between the total mean curvatures of two closed surfaces in ${\Bbb R}^3$ in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose is bound.Abstract:
In this paper we bound the difference between the total mean curvatures of two closed surfaces in ℝ3 in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.read more
Citations
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Proceedings ArticleDOI
Parallel Computation of Persistent Homology using the Blowup Complex
Ryan H. Lewis,Dmitriy Morozov +1 more
TL;DR: A parallel algorithm that computes persistent homology, an algebraic descriptor of a filtered topological space, by operating on a spatial decomposition of the domain, as opposed to a decomposition with respect to the filtration.
Journal ArticleDOI
Uniqueness of models in persistent homology: the case of curves
Patrizio Frosini,Claudia Landi +1 more
TL;DR: It is proved that f = g o h, where h: S1 ?
Journal ArticleDOI
Approximation and convergence of the intrinsic volume
TL;DR: In this article, a modification of the classic notion of intrinsic volume using persistence moments of height functions is introduced, which converges to the first intrinsic volume of the body as the resolution of the approximation improves.
Journal ArticleDOI
Stable Length Estimates of Tube-Like Shapes
TL;DR: An algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space using a variant of the Koksma–Hlawka Theorem and tools from computational topology to decrease the dependence on small perturbations of the shape.
Journal ArticleDOI
Uniqueness of models in persistent homology: the case of curves
Patrizio Frosini,Claudia Landi +1 more
TL;DR: In this paper, the persistent homology groups of a filtration induced on S 1 by f were analyzed for generic C1 functions and it was shown that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s?2, the persistent Betti number functions of s o f and s o g coincide, with respect to a suitable distance.
References
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Book
Differential geometry of curves and surfaces
TL;DR: This paper presents a meta-geometry of Surfaces: Isometrics Conformal Maps, which describes how the model derived from the Gauss Map changed over time to reflect the role of curvature in the model construction.
Book
Elements of Algebraic Topology
TL;DR: Elements of Algebraic Topology provides the most concrete approach to the subject with coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorem of point-set topology.
Book
Integral geometry and geometric probability
Abstract: Part I. Integral Geometry in the Plane: 1. Convex sets in the plane 2. Sets of points and Poisson processes in the plane 3. Sets of lines in the plane 4. Pairs of points and pairs of lines 5. Sets of strips in the plane 6. The group of motions in the plane: kinematic density 7. Fundamental formulas of Poincare and Blaschke 8. Lattices of figures Part II. General Integral Geometry: 9. Differential forms and Lie groups 10. Density and measure in homogenous spaces 11. The affine groups 12. The group of motions in En Part III. Integral Geometry in En: 13. Convex sets in En 14. Linear subspaces, convex sets and compact manifolds 15. The kinematic density in En 16. Geometric and statistical applications: stereology Part IV. Integral Geometry in Spaces of Constant Curvature: 17. Noneuclidean integral geometry 18. Crofton's formulas and the kinematic fundamental formula in noneuclidean spaces 19. Integral geometry and foliated spaces: trends in integral geometry.
Journal ArticleDOI
Topological Persistence and Simplification
Edelsbrunner,Letscher,Zomorodian +2 more
TL;DR: Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Book
Stratified Morse theory
Mark Goresky,Robert MacPherson +1 more
TL;DR: In this paper, the fundamental problem of Morse theory is to study the topological changes in the space X ≤c as the number c varies, where X is a topological space and c is a real number.