We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve. So far no algorithm was known for the calculation of such a shortest polygonal curve. This paper provides an iterative algorithmic solution, including a presentation of its foundations and of experimental results.

Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered that there was no royal road to geometry. iii Proclus Diadochus

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 a ”rubber-band algorithm” is known to compute such a curve approximately. We provide an alternative iterative algorithm for the approximative calculation of the MLP for curves contained in a special class of simple cube-curves (for which we prove the correctness of our alternative algorithm), and the obtained results coincide with those calculated by the rubber-band algorithm. 1

We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve. So far no algorithm was known for the calculation of such a shortest polygonal curve. This paper provides an iterative algorithmic solution, including a presentation of its foundations and of experimental results. 1.

Abstract. We consider simple cube-curves in the orthogonal 3D grid. 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 an MLP. A proof that this algorithm always converges to the correct curve, is still an open problem. This paper proves that the rubber-band algorithm is correct for the family of first-class simple cube-curves. 1

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) 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 [7] is related to the design of algorithms for the calculation of the MLP of a simple cube-curve: Is there a simple cubecurve 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. 1

One possible definition of the length of a digitized curve in 3D is the length of the shortest polygonal curve lying entirely in a cube curve. In earlier work the authors proposed an iterative algorithm for the calculation of this minimal length polygonal curve (MLP). This paper reviews the algorithm and suggests methods to speed it up by reducing the set of possible locations of vertices of the MLP, or by directly calculating MLP-vertices in specific situations. Altogether, the paper suggests an in-depth analysis of cube curves.

We consider simple cube-curves in the orthogonal 3D grid. 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 no provable general algorithm is known for the approximative calculation of such an MLP. This paper presents two approximative algorithms for computing the MLP of a general simple cube-curve in O(n 4) time, where n is the total number of critical edges of the given simple cube-curve.