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Topological Structures in Two‐Parameter‐Dependent 2D Vector Fields
Author(s) -
Weinkauf T.,
Theisel H.,
Hege H.C.,
Seidel H.P.
Publication year - 2006
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2006.00980.x
Subject(s) - euclidean vector , computer graphics , vector graphics , topology (electrical circuits) , computer science , topological data analysis , graphics , topological skeleton , line (geometry) , topological conjugacy , mathematics , algorithm , artificial intelligence , pure mathematics , geometry , computer graphics (images) , combinatorics , segmentation , active shape model
In this paper we extract and visualize the topological skeleton of two‐parameter‐dependent vector fields. This kind of vector data depends on two parameter dimensions, for instance physical time and a scale parameter. We show that two important classes of local bifurcations – fold and Hopf bifurcations – build line structures for which we present an approach to extract them. Furthermore we show that new kinds of structurally stable local bifurcations exist for this data, namely fold‐fold and Hopf‐fold bifurcations. We present a complete classification of them. We apply our topological extraction method to analyze a number of two‐parameter‐dependent vector fields with different physical interpretations of the two additional dimensions. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.3.3 [Computer Graphics]: Picture/Image Generation