Approximation of 3D Shortest Polygons in Simple Cube Curves

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. 1 Department of Computer and Information Science University of Pennsylvania, Philadelphia, USA 2 Center for Image Technology and Robotics Tamaki Campus, The University of Auckland, Auckland, New Zealand [email protected] You are granted permission for the non-commercial reproduction, distribution, display, and performance of this technical report in any format, BUT this permission is only for a period of 45 (forty-five) days from the most recent time that you verified that this technical report is still available from the CITR Tamaki web site under terms that include this permission. All other rights are reserved by the author(s). Approximation of 3D Shortest Polygons in Simple Cube Curves Thomas B ulow (1) and Reinhard Klette (2) (1) Department of Computer and Information Science University of Pennsylvania, Philadelphia, USA (2) CITR, University of Auckland, Tamaki Campus, Building 731 Auckland, New Zealand Abstract. One possible de nition 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 speci c situations. Altogether, the paper suggests an in-depth analysis of cube curves. One possible de nition 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 speci c situations. Altogether, the paper suggests an in-depth analysis of cube curves.