Translation Surfaces and Isotropic Transport Nets on Rational Minimal Surfaces

  • Jan VršekEmail author
  • Miroslav Lávička
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10521)


We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.



The authors were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. We thank to all referees for their valuable comments, which helped us to improve the paper.


  1. 1.
    Farin, G.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann Publishers Inc., San Francisco (2002)Google Scholar
  2. 2.
    Krivoshapko, S., Ivanov, V.: Encyclopedia of Analytical Surfaces. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  3. 3.
    Weinert, K., Du, S., Damm, P., Stautner, M.: Swept volume generation for the simulation of machining processes. Int. J. Mach. Tools Manuf. 44(6), 617–628 (2004)CrossRefGoogle Scholar
  4. 4.
    Shen, L.-Y., Pérez-Díaz, S.: Characterization of rational ruled surfaces. J. Symbolic Comput. 63, 21–45 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Pérez-Díaz, S., Shen, L.-Y.: Parametrization of translational surfaces. In: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, SNC 2014, New York, NY, USA, pp. 128–129. ACM (2014)Google Scholar
  6. 6.
    Vršek, J., Lávička, M.: Determining surfaces of revolution from their implicit equations. J. Comput. Appl. Math. 290, 125–135 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Vršek, J., Lávička, M.: Recognizing implicitly given rational canal surfaces. J. Symbolic Comput. 74, 367–377 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Struik, D.J.: Lectures on Classical Differential Geometry, 2nd edn. Dover Publications, Mineola (1988)zbMATHGoogle Scholar
  9. 9.
    Pottmann, H.: Rational curves and surfaces with rational offsets. Comput. Aided Geom. Des. 12(2), 175–192 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shafarevich, I.R.: Basic Algebraic Geometry. Springer, Heidelberg (1974)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mumford, D.: Algebraic Geometry I: Complex Projective Varieties. Springer, Heidelberg (1976)zbMATHGoogle Scholar
  12. 12.
    Odehnal, B.: On algebraic minimal surfaces. KoG 20, 61–78 (2016). Scientific and Professional Journal of Croatian Society for Geometry and GraphicsMathSciNetzbMATHGoogle Scholar
  13. 13.
    Weierstrass, K.: Über die flächen deren mittlere krümmung Überall gleich null ist. Monatsberichte der Berliner Akademie, pp. 612–625 (1866)Google Scholar
  14. 14.
    Schwarz, H.A.: Miscellen aus dem Gebiete der Minimalflächen, pp. 168–169. Springer, Heidelberg (1890)Google Scholar
  15. 15.
    Darboux, G.: Lecons sur la Theorie Generale des Surfaces et les Applications Geometriques Due Calcul Infinitesimal. Chelsea Publishing Series. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  16. 16.
    Jakob, R., Dierkes, U., Küster, A., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2010)Google Scholar
  17. 17.
    Osserman, R.: A Survey of Minimal Surfaces. Dover Publications, Mineola (2014)zbMATHGoogle Scholar
  18. 18.
    Lie, S.: Synthetischanalytische Untersuchungen über Minimalflächen. I. Über reelle algebraische Minimalflächen. Archiv for Mathematik og Naturvidenskab, pp. 157–198 (1877)Google Scholar
  19. 19.
    Geiser, C.F.: Notiz über die algebraischen Minimumsflächen. Math. Ann. 3(4), 530–534 (1871)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Farouki, R.T., Šír, Z.: Rational Pythagorean-hodograph space curves. Comput. Aided Geom. Des. 28, 75–88 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM J. Res. Develop. 34(5), 736–752 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Farouki, R.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  23. 23.
    Piegl, L., Tiller, W.: The NURBS Book. Monographs in Visual Communication, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lávička, M., Vršek, J.: On a special class of polynomial surfaces with Pythagorean normal vector fields. In: Boissonnat, J.-D., Chenin, P., Cohen, A., Gout, C., Lyche, T., Mazure, M.-L., Schumaker, L. (eds.) Curves and Surfaces 2010. LNCS, vol. 6920, pp. 431–444. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-27413-8_27 CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.NTIS – New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic

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