On-line construction of the convex hull of a simple polyline
Information Processing Letters
Approximate Euclidean shortest path in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Length Estimation in 3-D Using Cube Quantization
Journal of Mathematical Imaging and Vision
Digital Curves in 3D Space and a Linear-Time Length Estimation Algorithm
IEEE Transactions on Pattern Analysis and Machine Intelligence
Digital Geometry: Geometric Methods for Digital Picture Analysis
Digital Geometry: Geometric Methods for Digital Picture Analysis
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Minimum-Length polygon of a simple cube-curve in 3d space
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Automated DNA fragments recognition and sizing through AFM image processing
IEEE Transactions on Information Technology in Biomedicine
An approximation algorithm for computing minimum-length polygons in 3D images
ACCV'10 Proceedings of the 10th Asian conference on Computer vision - Volume Part IV
Approximate shortest paths in simple polyhedra
DGCI'11 Proceedings of the 16th IAPR international conference on Discrete geometry for computer imagery
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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) contained and complete in the tube of the curve. Only one general algorithm, called rubberband algorithm, was known for the approximative calculation of such an MLP so far. An open problem in [R. Klette and A. Rosenfeld. Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco, 2004.] is related to the design of algorithms for the calculation of the MLP of a simple cube-curve: Is there a simple cube-curve such that none of the nodes of its MLP is a grid vertex? This paper constructs an example of such a simple cube-curve, and we also characterize the class of all of such cube-curves. This study leads to a correction in Option 3 of the rubberband algorithm (by adding one missing test). We also prove that the rubberband algorithm has linear time complexity O(m) where m is the number of critical edges of a given simple cube-curve, which solves another open problem in the context of this algorithm.