Minimum-Length Polygons in Simple Cube-Curves

Simple cube-curves in a 3D orthogonal grid are polyhedrally bounded sets which 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. A critical edge is a grid edge contained in three consecutive cubes of a simple cube-curve. This paper shows that critical edges are the only possible locations of vertices of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve.