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Paths in Cube-Curves

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Book cover Euclidean Shortest Paths

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Abstract

This chapter discusses a problem defined in a 3-dimensional regular grid. Such a grid is commonly used in 3D image analysis. We may also assume that a general 3D space (e.g., for a robot) is regularly subdivided into cubes of uniform size. The chapter considers shortest paths in such a cuboidal world.

When I was a Boy Scout, we played a game when new Scouts joined the troop. We lined up chairs in a pattern, creating an obstacle course through which the new Scouts, blindfolded, were supposed to manoeuvre. The Scoutmaster gave them a few moments to study the pattern before our adventure began. But as soon as the victims were blindfolded, the rest of us quietly removed the chairs.—I think life is like this game.

Pierce Vincent Eckhart

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Notes

  1. 1.

    Published by Reinhard Klette and Thomas Bülow in 2000; see [32].

  2. 2.

    Published between 2000 and 2002 in papers by Thomas Bülow and Reinhard Klette; see [11].

  3. 3.

    See, for example, [17], pages 595–601.

  4. 4.

    Note that we can classify a simple cube-curve in linear time to be first-class or not, by using the original rubberband algorithm: the curve is first-class iff option (O 2) does not occur at all.

  5. 5.

    See, for example, [12, page 49].

  6. 6.

    See [12, page 49].

  7. 7.

    Formulated as an open problem on [33, page 406].

  8. 8.

    This leads to two new open problems (smallest simple cube-curve without end-angle, and smallest simple cube-curve where none of the MLP vertices is a grid point). See Problem 9.7 at the end of this chapter. We consider that the second problem (i.e., all MLP vertices not at a grid point) is more difficult to solve.

  9. 9.

    Note that it is impossible that four consecutive vertices of ρ are identical.

  10. 10.

    Otherwise, we update e by removing a sufficiently small segment(s) from its endpoint. This is another way to handle the degenerate case (see Sect. 3.4) of the used RBA.

  11. 11.

    For Line 2, see Lemma 6 in [40].

  12. 12.

    Two digits are used only for displaying coordinates. Obviously, in the calculations it is necessary to use higher precision.

  13. 13.

    This “hardness” is described in [45], page 666, or in [46].

  14. 14.

    A proof can be based on a theorem by C. Bajaj [6] and the factorisation algorithm by E.R. Berlekamp [7]. Details are given further below. Chandrajit Bajaj is with the University of Texas. Elwyn R. Berlekamp is with the University of California at Berkeley.

  15. 15.

    Theorem by E. Galois; see also B.L. van der Waerden’s famous example p(x)=x 5−x−1.

  16. 16.

    See Theorem 9 in [5], saying that the ESP problem is in general not solvable by radicals over the field of rationals.

  17. 17.

    The following results are well known in mathematical algebra; proofs can be found, for example, in [6, 24].

  18. 18.

    We follow [6]. However, this notion is not uniformly defined in literature; for a different use, see [62], for example.

  19. 19.

    Named after the Norwegian mathematician Niels Henrik Abel (1802–1829).

  20. 20.

    See, for example, [24] for more details. The following two lemmas are Theorems 5.7.1 and 5.7.2 in [24].

  21. 21.

    See, for example, [36].

  22. 22.

    See [36], page 441.

  23. 23.

    We used GAP, see [22].

  24. 24.

    For each a∈ℤ19, there exists a number j∈{0,1,…,18} such that 2j≡amod 19.

  25. 25.

    The DSS technique represents a digital curve, assumed to be a sequence of grid points, by subsequent digital straight segments (DSSs) of maximum length; the MLP technique considers a digital curve as a sequence of grid cells, calculating a minimum-length polygon (MLP) in the union of these cells.

  26. 26.

    Not to be confused with a 2D image segmentation algorithm of the same name [44].

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Authors and Affiliations

  1. School of Information Science and Technology, Huaqiao University, P.O. Box 800, Xiamen, Fujian, People’s Republic of China

    Fajie Li

  2. Dept. Computer Science, University of Auckland, P.O. Box 92019, Auckland, 1142, New Zealand

    Reinhard Klette

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  1. Fajie Li
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Li, F., Klette, R. (2011). Paths in Cube-Curves. In: Euclidean Shortest Paths. Springer, London. https://doi.org/10.1007/978-1-4471-2256-2_9

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  • DOI: https://doi.org/10.1007/978-1-4471-2256-2_9

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