Skip to main content
SpringerLink
Log in

Higher Dimensional Geometries. What Are They Good For?

  • Conference paper
  • First Online:
ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics (ICGG 2018)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 809))

Included in the following conference series:

  • 304 Accesses

Abstract

Geometries in higher dimensional spaces have many applications. We shall give a compilation of a few well-known examples here. The fact that some higher dimensional geometries can be found within some lower dimensional geometries makes them even more interesting. At hand of some familiar examples, we shall see what these concepts in geometry can do for us. In the beginning, the meaning of dimension will be clarified and an agreement is reached about what is higher dimensional. A few words will be said about the relations and interplay between models of various geometries. To the concept of model spaces a major part of this contribution will be dedicated to. A full section is dedicated to the applications of higher dimensional geometries.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 509.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 649.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berzolari, L., Rohn, K.: Algebraische Raumkurven und abwickelbare Flächen. Enzykl. Math. Wiss. Bd. 3-2-2a, B.G. Teubner, Leipzig, (1926)

    Google Scholar 

  2. Blaschke, W.: Vorlesungen über Differentialgeometrie III. Springer, Berlin (1929)

    MATH  Google Scholar 

  3. Burau, W.: Mehrdimensionale und höhere projektive Geometrie. VEB Deutscher Verlag der Wissenschaften, Berlin (1961)

    MATH  Google Scholar 

  4. Cecil, T.E.: Lie Sphere Geometry, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  5. Coolidge, J.L.: A Treatise on the Circle and the Sphere. Clarendon, Oxford (1916)

    MATH  Google Scholar 

  6. Cremona, L.: Elemente der Projektiven Geometrie. Verlag Cotta, Stuttgart (1882)

    MATH  Google Scholar 

  7. Giering, O.: Vorlesungen Über Höhere Geometrie. Vieweg, Braunschweig (1982)

    Book  Google Scholar 

  8. Glaeser, G., Stachel, H., Odehnal, B.: The Universe of Conics. From the ancient Greeks to 21st century developments. Springer-Verlag, Heidelberg (2016)

    Google Scholar 

  9. Graßmann, H.: Die Ausdehnungslehre. Verlag Th. Enslin, Berlin (1862)

    Google Scholar 

  10. Havlicek, H., Odehnal, B., Saniga, M.: Factor-group-generated polar spaces and (multi-)Qudits. SIGMA Symm. Integrab. Geom. Meth. Appl. 5/098, 15 pp (2009)

    Google Scholar 

  11. Havlicek, H., Kosiorek, J., Odehnal, B.: A point model for the free cyclic submodules over ternions. Results Math. 63, 1071–1078 (2013)

    Article  MathSciNet  Google Scholar 

  12. Havlicek, H., List, K., Zanella, C.: On automorphisms of flag spaces. Linear Multilinear Algebra 50, 241–251 (2002)

    Article  MathSciNet  Google Scholar 

  13. Hirschfeld, J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  14. Hofer, M., Odehnal, B., Pottmann, H., Steiner, T., Wallner, J.: 3D shape recognition and reconstruction based on line element geometry. In: 10th IEEE International Confirence Computer Vision, vol. 2, pp. 1532–1538. IEEE Computer Society, 2005, ISBN 0-7695-2334-X

    Google Scholar 

  15. Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A.K. Peters Ltd., Natick, MA (1993)

    MATH  Google Scholar 

  16. Klawitter, D.: Clifford Algebras. Geometric Modelling and Chain Geometries with Application in Kinematics. Ph.D. thesis, TU Dresden (2015)

    Google Scholar 

  17. Odehnal, B.: Flags in Euclidean three-space. Math. Pannon. 17(1), 29–48 (2006)

    MathSciNet  Google Scholar 

  18. Odehnal, B., Pottmann, H., Wallner, J.: Equiform kinematics and the geometry of line elements. Beitr. Algebra Geom. 47(2), 567–582 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Odehnal, B.: Die Linienelemente des \({\mathbb{P}}^3\). Österreich. Akad. Wiss. math.-naturw. Kl. S.-B. II(215), 155–171 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Odehnal, B.: Subdivision algorithms for ruled surfaces. J. Geom. Graphics 12(1), 35–52 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Odehnal, B.: Hermite interpolation with ruled and channel surfaces. G - slovenský Časopis pre Geometriu a Grafiku 14(28), 35–58 (2017)

    Google Scholar 

  22. Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Berlin - Heidelberg - New York (2001)

    MATH  Google Scholar 

  23. Pottmann, H., Hofer, M., Odehnal, B., Wallner, J.: Line geometry for 3D shape understanding and reconstruction. In: Pajdla, T., Matas, J. (eds.) Computer vision - ECCV 2004, Part I, volume 3021 of Lecture Notes in Computer Science, pp. 297–309. Springer, 2004, ISBN 3-540-21984-6

    Google Scholar 

  24. Segre, C.: Mehrdimensionale Räume. Enzykl. Math. Wiss. Bd. 3-2-2a, B.G. Teubner, Leipzig, (1912)

    Google Scholar 

  25. Study E.: Geometrie der Dynamen. B.G. Teubner, Leipzig (1903)

    Google Scholar 

  26. Veronese, G.: Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten geradliniger Einheiten, in elementarer Form entwickelt. B.G. Teubner, Leipzig (1894)

    MATH  Google Scholar 

  27. Wallner, J., Pottmann, H.: Intrinsic Subdivision with Smooth Limits for Ggraphics and Animation. ACM Trans. Graphics 25/2 (2006), 356–374

    Google Scholar 

  28. Weiss, E.A.: Einführung in die Liniengeometrie und Kinematik. B.G. Teubner, Leipzig (1935)

    MATH  Google Scholar 

  29. Windisch, G., Odehnal, B., Reimann, R., Anderhuber, F., Stachel, H.: Contact areas of the tibiotalar joint. J. Orthopedic Res. 25(11), 1481–1487 (2007)

    Article  Google Scholar 

  30. Zindler, K.: Liniengeometrie mit Anwendungen I. II. G.J. Göschen’sche Verlagshandlung, Leipzig (1906)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Applied Arts Vienna, Vienna, Austria

    Boris Odehnal

Authors
  1. Boris Odehnal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Odehnal .

Editor information

Editors and Affiliations

  1. Department of Architecture and Urban Studies—DASTU, School of Architecture Urban Planning and Construction Engineering—AUIC, Politecnico di Milano, Milan, Italy

    Luigi Cocchiarella

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Odehnal, B. (2019). Higher Dimensional Geometries. What Are They Good For?. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-95588-9_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95587-2

  • Online ISBN: 978-3-319-95588-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation