Generalized offsetting of planar structures using skeletons

We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams

Polygon subdivision for pocket machining process planning

In the process planning for pocket machining, selection of the optimal tool sizes and minimizing the number of plunging operations are among the most important factors in minimizing the machining time. This thesis presents a new approach for optimal tool selection of arbitrary shaped pockets based on a polygon subdivision technique. The pocket is subdivided to obtain smaller sub-polygons. The tools are selected separately for each sub-polygon and then the optimal set of the tools for the entire pocket is obtained based on minimizing both the machining time and the number of tools used to machine the pocket. Finally, the sub-polygons are sequenced in an optimal order to eliminate the requirement of multiple plunging operations. The approach presented is an improvement over previous work because it makes an effective use of the polygon subdivision strategy to improve the machining time as well as reducing the number of plunges. The implementation examples of this approach suggest that the machining time can be improved as much as 75%

Parallel Contour Path Planning for Complicated Cavity Part Fabrication using Voronoi-based Distance Map

To generate parallel contour path for direct production of complicated cavity component, a novel path planning based on Voronoi-based distance map is presented in this paper. Firstly, the grid representation of polygonal slice is produced by hierarchical rasterization using graphics hardware acceleration and divided into Voronoi cells of contour by an exact EDT (Euclidean distance transformation). Then, each VCI (Voronoi cell of inner contour) is further subdivided into CLRI (closed loop region of inner contour) and OLRI (open loop region of inner contour). Closed paths for each CLRI and the block merging VCO (Voronoi cell of outer contour) and all OLRIs are generated by local and global isoline extraction, respectively. The final path ordered in circumferential and radial directions is obtained by sorting and connecting all individual paths. In comparison with conventional methods such as pair-wise intersection and Voronoi diagram, the proposed algorithm is numerically robust, can avoid null path and self-intersection because of the application of distance map and discrete Voronoi diagram. It is especially used for FGM (Functionally Graded Material) design and fabrication.Mechanical Engineerin

Finding the Maximum Subset with Bounded Convex Curvature

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When machining a pocket in a solid piece of material such as steel using a rough tool in a milling machine, sharp convex corners of the pocket cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a tool path that maximizes the use of the rough tool. Mathematically, this boils down to the following problem. Given a simply-connected set of points P in the plane such that the boundary of P is a curvilinear polygon consisting of n line segments and circular arcs of arbitrary radii, compute the maximum subset Q of P consisting of simply-connected sets where the boundary of each set is a curve with bounded convex curvature. A closed curve has bounded convex curvature if, when traversed in counterclockwise direction, it turns to the left with curvature at most 1. There is no bound on the curvature where it turns to the right. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of the pocket. We devise an algorithm to compute the unique maximum such set Q. The algorithm runs in O(n log n) time and uses O(n) space. For the correctness of our algorithm, we prove a new generalization of the Pestov-Ionin Theorem. This is needed to show that the output Q of our algorithm is indeed maximum in the sense that if Q\u27 is any subset of P with a boundary of bounded convex curvature, then Q\u27 is a subset of Q

A unified rough and finish cut algorithm for NC machining of free form pockets using a grid based approach

http://www.worldcat.org/oclc/4333435

A Systematic Review of Algorithms with Linear-time Behaviour to Generate Delaunay and Voronoi Tessellations

Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented

Path Planning for Mobile Robot Navigation using Voronoi Diagram and Fast Marching

For navigation in complex environments, a robot need s to reach a compromise between the need for having efficient and optimized trajectories and t he need for reacting to unexpected events. This paper presents a new sensor-based Path Planner w hich results in a fast local or global motion planning able to incorporate the new obstacle information. In the first step the safest areas in the environment are extracted by means of a Vorono i Diagram. In the second step the Fast Marching Method is applied to the Voronoi extracted a reas in order to obtain the path. The method combines map-based and sensor-based planning o perations to provide a reliable motion plan, while it operates at the sensor frequency. The m ain characteristics are speed and reliability, since the map dimensions are reduced to an almost uni dimensional map and this map represents the safest areas in the environment for moving the robot. In addition, the Voronoi Diagram can be calculated in open areas, and with all kind of shaped obstacles, which allows to apply the proposed planning method in complex environments wher e other methods of planning based on Voronoi do not work.This work has been supported by the CAM Project S2009/DPI-1559/ROBOCITY2030 I

Finding Wombling Boundaries in LHC Data with Voronoi and Delaunay Tessellations

We address the problem of finding a wombling boundary in point data generated by a general Poisson point process, a specific example of which is an LHC event sample distributed in the phase space of a final state signature, with the wombling boundary created by some new physics. We discuss the use of Voronoi and Delaunay tessellations of the point data for estimating the local gradients and investigate methods for sharpening the boundaries by reducing the statistical noise. The outcome from traditional wombling algorithms is a set of boundary cell candidates with relatively large gradients, whose spatial properties must then be scrutinized in order to construct the boundary and evaluate its significance. Here we propose an alternative approach where we simultaneously form and evaluate the significance of all possible boundaries in terms of the total gradient flux. We illustrate our method with several toy examples of both straight and curved boundaries with varying amounts of signal present in the data.Comment: 54 pages. New figure 13 and appendix A added. Conclusions unchanged. Matches published versio