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Graphs and Combinatorics

, Volume 27, Issue 3, pp 451–463 | Cite as

Classification of the Congruent Embeddings of a Tetrahedron into a Triangular Prism

  • H. MaeharaEmail author
  • N. Tokushige
Proceedings Paper

Abstract

Let P(t) denote an infinitely long right triangular prism whose base is an equilateral triangle of edge length t. Let \({{\mathcal F}(t)}\) be the family of those subsets of P(t) that are congruent to a regular tetrahedron of unit edge. We present complete classification of the members of \({{\mathcal F}(t)}\) modulo rigid motions within the prism P(t), for every t > 0.

Keywords

Embeddings of a tetrahedron Triangular prism Classification 

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References

  1. 1.
    Bárány, I., Maehara, H., Tokushige, N.: Tetrahedra passing through a triangular hole. (2011, submitted)Google Scholar
  2. 2.
    Brandenberg R., Theobald T.: Radii minimal projections of polytopes and constrained optimization of symmetric polynomials. Adv. Geom. 6, 71–83 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brandenberg R., Theobald T.: Algebraic method for computing smallest enclosing and circumscribing cylinders of simplices. Appl. Algebra Eng. Comm. Comput. 14, 439–460 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Devillers O., Mourrain B., Preparata F.P., Trebuchet P.: Circular cylinders through four or five points in space. Discrete Comput. Geom. 29, 83–104 (2003)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Itoh J., Tanoue Y., Zamfirescu T.: Tetrahedra passing through a circular or square hole. Rend. Circ. Mat. Palermo (2) Suppl. 77, 349–354 (2006)MathSciNetGoogle Scholar
  6. 6.
    Maehara H.: An extremal problem for arrangements of great circles. Math. Japonica 41, 125–129 (1995)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Maehara H.: On congruent embeddings of a tetrehedron into a circular cylinder. Yokohama Math. J. 55, 171–177 (2010)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Pukhov V.: Kolmogorov diameter of a regular simplex. Mosc. Univ. Math. Bull. 35, 38–41 (1980)zbMATHGoogle Scholar
  9. 9.
    Schömer E., Sellen J., Reichmann M., Yap C.: Smallest enclosing cylinders. Algorithmica 27, 170–186 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Steinhagen P.: Über die grösste Kugel in einer konvexen Punktmenge. Abh. Math. Sem. Hamburg 1, 15–26 (1921)CrossRefGoogle Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Research Institute of Educational DevelopmentTokai UniversityTokyoJapan
  2. 2.College of EducationRyukyu UniversityNishihara, OkinawaJapan

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