Visually Evaluating the Topological Equivalence of Bounded Bivariate Fields

Daisuke Sakurai, Takahiro Yamamoto

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We apply visualization to evaluating a new topological equivalence relation, which we call the topological B+ -equivalence. It has been used in our separate, yet ongoing, study in mathematics. The equivalence is a building block for the topological study of maps of bounded manifolds into the plane (aka bounded bivariate fields). In that study, we have introduced a few invariants that approximate the equivalence, which is hard to treat directly. In this chapter dedicated to the visualization community, we show that visualizing the Reeb space gives us a near-instant way of evaluating the invariants. The process has traditionally required an unpredictable amount of time due to manual analysis of high-order polynomials, which was necessary to obtain the invariant values. Our Reeb space visualization reveals the topological information necessary for evaluating the invariants, and, in doing so, the topological B+ -equivalence itself. Previously, the visualization was found to serve as an introductory learning tool for studying examples of singular fibers. The present article goes further to demonstrate professional use cases.

Original languageEnglish
Title of host publicationTopological Methods in Data Analysis and Visualization VI - Theory, Applications, and Software
EditorsIngrid Hotz, Talha Bin Masood, Filip Sadlo, Julien Tierny
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages16
ISBN (Print)9783030834999
Publication statusPublished - 2021
Event8th Workshop on Topological Methods in Data Analysis and Visualization, TopoInVis 2019 - Nyköping, Sweden
Duration: Jun 17 2019Jun 19 2019

Publication series

NameMathematics and Visualization
ISSN (Print)1612-3786
ISSN (Electronic)2197-666X


Conference8th Workshop on Topological Methods in Data Analysis and Visualization, TopoInVis 2019

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Geometry and Topology
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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