TY - GEN
T1 - Visually Evaluating the Topological Equivalence of Bounded Bivariate Fields
AU - Sakurai, Daisuke
AU - Yamamoto, Takahiro
N1 - Funding Information:
Acknowledgment This work was supported by JSPS KAKENHI (Grant Number JP20K19809) and IMI Joint Use Research Program Workshop (II) 20200011 “Fiber Topology Meets Applications.”
Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85116728500&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85116728500&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-83500-2_10
DO - 10.1007/978-3-030-83500-2_10
M3 - Conference contribution
AN - SCOPUS:85116728500
SN - 9783030834999
T3 - Mathematics and Visualization
SP - 181
EP - 196
BT - Topological Methods in Data Analysis and Visualization VI - Theory, Applications, and Software
A2 - Hotz, Ingrid
A2 - Bin Masood, Talha
A2 - Sadlo, Filip
A2 - Tierny, Julien
PB - Springer Science and Business Media Deutschland GmbH
T2 - 8th Workshop on Topological Methods in Data Analysis and Visualization, TopoInVis 2019
Y2 - 17 June 2019 through 19 June 2019
ER -