Skip to main content

Advertisement

Springer Nature Link
Log in

A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices

  • Regular Paper
  • Published:
Journal of Visualization Aims and scope Submit manuscript

Abstract

A method to visualize polytopes in a four-dimensional euclidian space (x, y, z, w) is proposed. A polytope is sliced by multiple hyperplanes that are parallel to each other and separated by uniform intervals. Since the hyperplanes are perpendicular to the w-axis, the resulting multiple slices appear in the three-dimensional (x, y, z) space and they are shown by the standard computer graphics. The polytope is rotated extrinsically in the four-dimensional space by means of a simple input method based on keyboard typings. The multiple slices are placed on a parabola curve in the three-dimensional world coordinates. The slices in a view window form an oval appearance. Both the simple and the double rotations in the four-dimensional space are applied to the polytope. All slices synchronously change their shapes when a rotation is applied to the polytope. The compact display in the oval of many slices with the help of quick rotations facilitate a grasp of the four-dimensional configuration of the polytope.

Graphical Abstract

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

Similar content being viewed by others

References

  • Abbott E (1992) Flatland: a romance of many dimensions. Dover Publications, New York

    MATH  Google Scholar 

  • Banks D (1992) Interactive manipulation and display of surfaces in four dimensions. In: Proceedings of the 1992 symposium on interactive 3D graphics, pp 197–207

  • Chu A, Fu CW, Hanson AJ, Heng PA (2009) GL4D: a GPU-based architecture for interactive 4D visualization. IEEE Trans Vis Comput Graph 15(6):1587–1594

    Article  Google Scholar 

  • Cole FN (1890) On rotations in space of four dimensions. Am J Math 12(2):191–210

    Article  MathSciNet  MATH  Google Scholar 

  • Hanson AJ, Cross RA (1993) Interactive visualization methods for four dimensions. In: Proceedings of visualization ’93, pp 196–203

  • Hanson AJ, Heng PA (1991) Visualizing the fourth dimension using geometry and light. In: Proceedings of the 2nd conference on visualization ’91, pp 321–328

  • Hanson AJ, Heng PA (1992) Four-dimensional views of 3D scalar fields. In: Proceedings of visualization ’92, pp 84–91

  • Hanson AJ, Ishkov KI, Ma JH (1999) Meshview: visualizing the fourth dimension. Technical report. http://www.cs.indiana.edu/%7Ehansona/meshview/meshview

  • Hausmann B, Seidel HP (1994) Visualization of regular polytopes in three and four dimensions. Comput Graph Forum 13(3):305–316

    Article  Google Scholar 

  • Hoffmann CM, Zhou J (1991) Some techniques for visualizing surfaces in four-dimensional space. Comput-Aided Des 23(1):83–91

    Article  MATH  Google Scholar 

  • Kageyama A (2015) Keyboard-bsed control of four-dimensional rotations. J Vis. doi:10.1007/s12650-015-0313-y

    Google Scholar 

  • Manning HP (1914) Geometry of four dimensions. MacMillan, New York

    MATH  Google Scholar 

  • Noll AM (1968) Computer animation and the fourth dimension. In: AFIPS conference proceedings, vol 33, 1968 Fall Joint Computer Conference, Thompson Book Company, Washington D.C., pp 1279–1283

  • Sakai Y, Hashimoto S (2011) Four-dimensional mathematical data visualization via “Embodied Four-dimensional Space Display System”. Forma 26(1):11–18

    MathSciNet  Google Scholar 

  • Sequin CH (2002) 3D Visualization models of the regular polytopes in four and higher dimensions. In: Proceedings of bridges: mathematical connections in art, music, and science, pp 37–48

  • Sullivan JM (1991) Generating and rendering four-dimensional polytopes. Math J 1:76–85

    Google Scholar 

  • van Wijk JJ, van Liere R (1993) HyperSlice. In: Proceedings of the 4th conference on visualization ’93, pp 119–125

  • Woodring J, Wang C, Shen HWW (2003) High dimensional direct rendering of time-varying volumetric data. Proc VIS 2003:417–424

    Google Scholar 

Download references

Acknowledgments

This work was supported by JSPS KAKENHI Grant No. 20260052. The author thanks a reviewer for pointing out the importance of subsidiary data in the slice visualization. The colored parallel coordinates are added by the reviewer’s suggestion.

Author information

Authors and Affiliations

  1. Department of Computational Science, Kobe University, Kobe, 657-8501, Japan

    Akira Kageyama

Authors
  1. Akira Kageyama
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Akira Kageyama.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kageyama, A. A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices. J Vis 19, 417–422 (2016). https://doi.org/10.1007/s12650-015-0319-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12650-015-0319-5

Keywords

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy