Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Relaxing topological surfaces in four dimensions

  • Original article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

In this paper, we show the use of visualization and topological relaxation methods to analyze and understand the underlying structure of mathematical surfaces embedded in 4D. When projected from 4D to 3D space, mathematical surfaces often twist, turn, and fold back on themselves, leaving their underlying structures behind their 3D figures. Our approach combines computer graphics, relaxation algorithm, and simulation to facilitate the modeling and depiction of 4D surfaces, and their deformation toward the simplified representations. For our principal test case of surfaces in 4D, this for the first time permits us to visualize a set of well-known topological phenomena beyond 3D that otherwise could only exist in the mathematician’s mind. Understanding a fairly long mathematical deformation sequence can be aided by visual analysis and comparison over the identified “key moments” where only critical changes occur in the sequence. Our interface is designed to summarize the deformation sequence with a significantly reduced number of visual frames. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical surfaces and their deformation in topological space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Abbott, E.A.: Flatland. Dover Publications, Inc., Mineola (1952)

    Google Scholar 

  2. Annas, J.: An Introduction to Plato’s Republic. Clarendon Press, Oxford (1981)

    Google Scholar 

  3. Artin, E.: Zur isotopie zweidimensionaler flächen im r 4. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 4, pp. 174–177. Springer (1925)

  4. Brown, R.: Mathematics and knots. In: Visual Representations and Interpretations, pp. 32–42. Springer (1999)

  5. Bruß, I., Frick, A.: Fast interactive 3-d graph visualization. In: International Symposium on Graph Drawing, pp. 99–110. Springer (1995)

  6. Carter, J.: How Surfaces Intersect in Space: An Introduction to Topology. K & E Series on Knots and Everything. World Scientific, Singapore (1995)

    Book  Google Scholar 

  7. Carter, J.S.: Reidemeister/roseman-type moves to embedded foams in 4-dimensional space (2012)

  8. Carter, J.S., Saito, M.: Knot diagrams and braid theories in dimension 4. Pitman Research Notes in Mathematics Series, pp. 112–112 (1995)

  9. Chetverikov, D., Stepanov, D., Krsek, P.: Robust euclidean alignment of 3d point sets: the trimmed iterative closest point algorithm. Image Vis. Comput. 23(3), 299–309 (2005)

    Article  Google Scholar 

  10. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Algorithms for drawing graphs: an annotated bibliography. Comput. Geom. Theory Appl. 4(5), 235–282 (1994)

    Article  MathSciNet  Google Scholar 

  11. Dutagaci, H., Cheung, C.P., Godil, A.: A benchmark for best view selection of 3d objects. In: Proceedings of the ACM Workshop on 3D Object Retrieval, 3DOR ’10, pp. 45–50. ACM, New York, NY, USA (2010)

  12. Eberly, D.H.: 3D game engine design: a practical approach to real-time computer graphics. CRC Press, Boca Raton (2006)

    Book  Google Scholar 

  13. Fitzgibbon, A.W.: Robust registration of 2d and 3d point sets. Image Vis. Comput. 21(13–14), 1145–1153 (2003)

    Article  Google Scholar 

  14. Fox, R.H.: A quick trip through knot theory. Topology 3, 120–167 (1962)

    MathSciNet  MATH  Google Scholar 

  15. Francis, G.K.: A Topological Picturebook. Springer, Berlin (1987)

    MATH  Google Scholar 

  16. Friedman, G.: Knot spinning, pp. 187–208. Handbook of knot theory. Elsevier Science, Amsterdam (2005)

    MATH  Google Scholar 

  17. Fruchterman, T.M., Reingold, E.M.: Graph drawing by force-directed placement. Softw.: Pract. Exp. 21(11), 1129–1164 (1991)

    Google Scholar 

  18. Hanson, A.J., Munzner, T., Francis, G.: Interactive methods for visualizable geometry. Computer 27(7), 73–83 (1994)

    Article  Google Scholar 

  19. Hass, J., Lagarias, J.C., Pippenger, N.: The computational complexity of knot and link problems. J. ACM (JACM) 46(2), 185–211 (1999)

    Article  MathSciNet  Google Scholar 

  20. J. Scott, Carter, M.: Knotted Surfaces and Their Diagrams. American Mathematical Soc, Providence (1998)

  21. Jordan, K., Miller, L.E., Moore, E., Peters, T., Russell, A.: Modeling time and topology for animation and visualization with examples on parametric geometry. Theor. Comput. Sci. 405(1), 41–49 (2008)

    Article  MathSciNet  Google Scholar 

  22. Kawauchi, A.: A survey of knot theory. Birkhäuser, Basel (2012)

    Google Scholar 

  23. Kobourov, S.G.: Spring embedders and force directed graph drawing algorithms. arXiv preprint arXiv:1201.3011 (2012)

  24. Kumar, A., Fowler, R.H.: A spring modeling algorithm to position nodes of an undirected graph in three dimensions. Tech. rep., Technical Report CS-94-7 (1994)

  25. Li, C., Sun, Z., Song, M., Zhang, Y.: Best view selection of 3d models based on unsupervised feature learning and discrimination ability. In: Proceedings of the 6th International Symposium on Visual Information Communication and Interaction, VINCI ’13, pp. 107–108. ACM, New York, NY, USA (2013)

  26. Livingston, C.: Knot Theory, The Carus Mathematical Monographs, vol. 24. Mathematical Association of America, Washington (1993)

    Google Scholar 

  27. Paton, R., Neilsen, I.: Visual representations and interpretations. Springer, Berlin (2012)

    Google Scholar 

  28. Ranicki, A.: High-Dimensional Knot Theory: Algebraic Surgery in Codimension 2. Springer, Berlin (2013)

    MATH  Google Scholar 

  29. Roseman, D.: Twisting and turning in four dimensions. Video animation, Department of Mathematics, University of Iowa, and the Geometry Center (1993)

  30. Scharein, R.G.: Interactive topological drawing. Ph.D. thesis, Department of Computer Science, The University of British Columbia (1998)

  31. Simmons, G.F., Hammitt, J.K.: Introduction to Topology and Modern Analysis. McGraw-Hill, New York (1963)

    Google Scholar 

  32. SIMON, J.K.: Energy functions for polygonal knots. J. Knot Theory Ramific. 03(03), 299–320 (1994)

    Article  MathSciNet  Google Scholar 

  33. Sorkine, O.: Least-squares rigid motion using SVD. Tech. Not. 120(3), 52 (2009)

    Google Scholar 

  34. Vázquez, P.P., Feixas, M., Sbert, M., Heidrich, W.: Viewpoint selection using viewpoint entropy. In: VMV ’01: Proceedings of the Vision Modeling and Visualization Conference 2001, pp. 273–280. Aka GmbH (2001)

  35. Vázquez, P.P., Sbert, M.: Fast adaptive selection of best views. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) Computational Science and Its Applications – ICCSA 2003, pp. 295–305. Springer, Berlin (2003)

    Chapter  Google Scholar 

  36. Wu, Y.: An md knot energy minimizing program. Department of Mathematics, University of Iowa

  37. Zhang, H., Hanson, A.: Shadow-driven 4d haptic visualization. IEEE Trans. Vis. Comput. Gr. 13(6), 1688–1695 (2007)

    Article  Google Scholar 

  38. Zhang, H., Hanson, A.J.: Physically interacting with four dimensions. In: G. Bebis, R. Boyle, B. Parvin, D. Koracin, P. Remagnino, A.V. Nefian, M. Gopi, V. Pascucci, J. Zara, J. Molineros, H. Theisel, T. Malzbender (eds.) ISVC (1), Lecture Notes in Computer Science, vol. 4291, pp. 232–242. Springer (2006)

  39. Zhang, H., Weng, J., Hanson, A.: A pseudo-haptic knot diagram interface. In: Proc. SPIE 7868, vol. 786807, pp. 1–14 (2011)

  40. Zhang, H., Weng, J., Jing, L., Zhong, Y.: Knotpad: visualizing and exploring knot theory with fluid reidemeister moves. IEEE Trans. Vis. Comput. Gr. 18(12), 2051–2060 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by National Science Foundation Grant #1651581 and the 2016 ORAU’s Ralph E. Powe Junior Faculty Enhancement grant.

Author information

Authors and Affiliations

  1. University of Louisville, Louisville, USA

    Hui Zhang & Huan Liu

Authors
  1. Hui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Liu, H. Relaxing topological surfaces in four dimensions. Vis Comput 36, 2341–2353 (2020). https://doi.org/10.1007/s00371-020-01895-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-020-01895-5

Keywords

Search

Navigation