The Class of Simple Cube-Curves Whose MLPs Cannot Have Vertices at Grid Points

  • Fajie Li
  • Reinhard Klette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)


We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve. So far only one general algorithm called rubber-band algorithm was known for the approximative calculation of such a MLP. There is an open problem which is related to the design of algorithms for calculation a 3D MLP of a cube-curve: Is there a simple cube-curve such that none of the vertices of its 3D MLP is a grid vertex? This paper constructs an example of such a simple cube-curve. We also characterize this class of cube-curves.


Critical Line Parallel Projection Simple Cube Grid Plane Robot Motion Planning 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Fajie Li
    • 1
  • Reinhard Klette
    • 1
  1. 1.CITRUniversity of AucklandAucklandNew Zealand

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