Abstract
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.
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Li, F., Klette, R. (2005). The Class of Simple Cube-Curves Whose MLPs Cannot Have Vertices at Grid Points. In: Andres, E., Damiand, G., Lienhardt, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2005. Lecture Notes in Computer Science, vol 3429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31965-8_18
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DOI: https://doi.org/10.1007/978-3-540-31965-8_18
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