The Class of Simple Cube-Curves Whose MLPs Cannot Have Vertices at Grid Points
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.
KeywordsCritical Line Parallel Projection Simple Cube Grid Plane Robot Motion Planning
- 3.Sloboda, F., Zaťko, B., Klette, R.: On the topology of grid continua. In: SPIE Vision Geometry VII, vol. 3454, pp. 52–63 (1998)Google Scholar
- 4.Sloboda, F., Zaťko, B., Stoer, J.: On approximation of planar one-dimensional grid continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160. Springer, Singapore (1998)Google Scholar
- 7.Canny, J., Reif, J.H.: New lower bound techniques for robot motion planning problems. In: Proc. IEEE Conf. Foundations Computer Science, pp. 49–60 (1987)Google Scholar
- 8.Choi, J., Sellen, J., Yap, C.-K.: Approximate Euclidean shortest path in 3-space. In: Proc. ACM Conf. Computational Geometry, pp. 41–48. ACM Press, New York (1994)Google Scholar