When the medial axis meets the singularities

Denkowski, Maciej Piotr

dc.contributor.author	Denkowski, Maciej Piotr
dc.contributor.editor	Krasiński, Tadeusz
dc.contributor.editor	Spodzieja, Stanisław
dc.date.accessioned	2020-01-24T07:37:42Z
dc.date.available	2020-01-24T07:37:42Z
dc.date.issued	2019
dc.identifier.citation	Denkowski M. P., When the medial axis meets the singularities, in: Analytic and Algebraic Geometry 3, T. Krasiński, S. Spodzieja (red.), WUŁ, Łódź 2019, doi: 10.18778/8142-814-9.05.	pl_PL
dc.identifier.isbn	978-83-8142-814-9
dc.identifier.uri	http://hdl.handle.net/11089/31258
dc.description.abstract	In this survey we present recent results in the study of the medial axes of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis – i.e. the set of points with more than one closest point to a given closed set X C Rn (with respect to the Euclidean distance) – reaches some singular points of X bringing along some metric information about them.	pl_PL
dc.language.iso	en	pl_PL
dc.publisher	Wydawnictwo Uniwersytetu Łódzkiego	pl_PL
dc.relation.ispartof	Analytic and Algebraic Geometry 3;
dc.rights	Attribution-NonCommercial-NoDerivatives 4.0 Międzynarodowe	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/4.0/	*
dc.title	When the medial axis meets the singularities	pl_PL
dc.type	Book chapter	pl_PL
dc.page.number	41-66	pl_PL
dc.contributor.authorAffiliation	Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics	pl_PL
dc.identifier.eisbn	978-83-8142-815-6
dc.references	D. Attali, J.-D. Boissonnat, H. Edelsbrunner, Stability and Computation of Medial Axes \| a State-of-the-Art Report, Mathematical Foundations of Scienti c Visualization, Computer Graphics, and Massive Data Exploration, T. M oller and B. Hamann and R. Russell (Ed.) (2009) 109-125;	pl_PL
dc.references	P. Albano, P. Cannarsa, Structural properties of singularities of semiconcave functions, Annali Scuola Norm. Sup. Pisa Sci. Fis. Mat. 28 (1999), 719-740;	pl_PL
dc.references	A. Bia lo_zyt, On the medial axis in singularity theory, PhD Thesis, Jagiellonian University, Krak ow 2019, in preparation;	pl_PL
dc.references	L. Birbrair, D. Siersma, Metric properties of con ict sets, Houston Journ. Math. 35 no. 1 (2009), 73-80;	pl_PL
dc.references	L. Birbrair, M. P. Denkowski, Medial axis and singularities, J. Geom. Anal. 27 no. 3 (2017), 2339-2380;	pl_PL
dc.references	L. Birbrair, M. P. Denkowski, Erratum to: Medial axis and singularities, arXiv:1705.02788 (2017);	pl_PL
dc.references	H. Blum, A transformation for extracting new descriptors of shape, in: Models for the perception of speech and visual form, W. Wathen-Dunn, ed. MIT Press, Cambridge, MA, 1967, 362-380;	pl_PL
dc.references	J. Bochnak, J.-J. Risler, Sur les exposants de Lojasiewicz, Comment. Math. Helv. 50 (1975), 493-508;	pl_PL
dc.references	F. Chazal i R. Sou et Stability and niteness properties of medial axis and skeleton, J. Dynam. Control Systems 10.2 (2004), 149-170;	pl_PL
dc.references	F. Clarke, Generalized gradients and applications, Trans. A. M. S. 205 (1975), 247-262;	pl_PL
dc.references	M. Coste, An introduction to o-minimal geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligra ci Internazionali, Pisa (2000);	pl_PL
dc.references	J. Damon, Swept regions and surfaces: modeling and volumetric properties, Theoret. Comput. Sci. 392, no. 1-3, (2008), 66-91;	pl_PL
dc.references	Z. Denkowska, M. P. Denkowski, The Kuratowski convergence and connected components, J. Math. Anal. Appl. (2012) 387 no. 1, 48-65;	pl_PL
dc.references	Z. Denkowska, M. P. Denkowski, A long and winding road to o-minimal structures, J. Sing. 13 (2015), 57-86,	pl_PL
dc.references	Z. Denkowska, J. Stasica, Ensembles sous-analytiques a la polonaise, Hermann 2007;	pl_PL
dc.references	M. P. Denkowski, On the points realizing the distance to a de nable set, J. Math. Anal. Appl. 378 no. 2 (2011), pp. 592-602;	pl_PL
dc.references	M. P. Denkowski, Sur la connexit e de l'axe m edian, Univ. Iagell. Acta Math. vol. 53 (2016), 7-12;	pl_PL
dc.references	M. P. Denkowski, The Kuratowski convergence of medial axes arXiv:1602.05422 (2016);	pl_PL
dc.references	M. P. Denkowski, On Yomdin's version of a Lipschitz Implicit Function Theorem, arXiv:1610.07905 (2016);	pl_PL
dc.references	M. P. Denkowski, P. Pe lszy nska, On de nable multifunctions and Lojasiewicz inequalities, J. Math. Anal. Appl. 456 no. 2 (2017), 1101-1122;	pl_PL
dc.references	D. H. Fremlin, Skeletons and central sets, Bull. London Math. Soc. (3) 74 (1997), 701-720;	pl_PL
dc.references	M. Ghomi, R. Howard, Tangent cones and regularity of hypersurfaces, J. Reine Angew. Math. 697 (2014), 221-247;	pl_PL
dc.references	A. Lieutier, Any open bounded subset of Rn has the same homotopy type as its medial axis, Computer-Aided Design 36 (2004) 10291046;	pl_PL
dc.references	D. Milman, The central function of the boundary of a domain and its di erentiable proper- ties, Journal of Geometry, Vol. 14/2 (1980), 182-202;	pl_PL
dc.references	D. Milman, Z. Waksman, On topological properties of the central set of a bounded domain in Rm, Journal of Geometry, Vol. 15/1 (1981), 1-7;	pl_PL
dc.references	J. Nash, Real algebraic manifolds, Ann. Math. 56 (1952), 405-421	pl_PL
dc.references	J.-B. Poly, G. Raby, Fonction distance et singularit es, Bull. Sci. Math. 2 eme s erie, 108 (1984), 187-195;	pl_PL
dc.references	Y. Yomdin, On the local structure of a generic central set, Comp. Math. 43 no. 2 (1981), 225-238.	pl_PL
dc.identifier.doi	10.18778/8142-814-9.05