Visibility from a Slice File for Rapid CNC Machining

A methodology for using CNC machining as a rapid prototyping process is being developed. The method involves cutting complex parts using layer-based machining operations from a plurality of orientations about one axis of rotation. A critical step is to determine the number of and location of those orientations. This paper presents an approach to mapping the visibility of a model about an axis of rotation using a set of model slices taken orthogonal to the axis of rotation

Determining Setup Orientations From the Visibility of Slice Geometry for Rapid Computer Numerically Controlled Machining

A method for rapid computer numerically controlled (CNC) machining is being developed in an effort to automatically create functional prototypes and parts in a wide array of materials. The method uses a plurality of simple two-and-a-half-dimensional (21/2-D) toolpaths from various orientations about an axis of rotation in order to machine the entire surface of a part without refixturing. It is our goal to automatically create these toolpaths for machining and eliminate the complex planning traditionally associated with CNC machining. In this paper, we consider a problem that arises in automating this process - visibility to the surface of a model that is rotated about a fourth axis. Our approach involves slicing the computer-aided design (CAD) model orthogonal to the axis of rotation. The slice geometry is used to calculate two-dimensional visibility maps for the set of polygons on each slice plane. The visibility data provides critical information for determining the minimum number and orientation of 21/2-D toolpaths required to machine the entire surface of a part

Detours admitting short paths

Finding shortest paths between two vertices in a weighted graph is a well explored problem and several efficient algorithms for solving it have been reported. We propose a new variation of this problem which we call the Detour Admitting Shortest Path Problem (DASPP).We present an efficient algorithm for solving DASPP. This is the first algorithm that constructs a shortest path such that each edge of the shortest path admits a detour with no more than k−hops. This algorithm has important applications in transportation networks. We also present implementation issues for the detour admitting shortest path algorithm

Watchman routes in the presence of convex obstacles

This thesis deals with the problem of computing shortest watchman routes in the presence of polygonal obstacles. Important recent results on watchman route problems are surveyed. An {dollar}O(n\sp3){dollar} algorithm for computing a shortest watchman route in the presence of a pair of convex obstacles is presented. Important open problems related to watchman route problems are discussed

Local-Global Results on Discrete Structures

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial implications of Bakry- Émery curvature on graphs to prove a sort of local discrepancy inequality. This then allows us to derive a number of results regarding the local structure of graphs, dependent only on a curvature lower bound. For instance, it turns out that a curvature lower bound implies a nontrivial lower bound on graph connectivity. We also use these results to consider the curvature of strongly regular graphs, a well studied and important class of graphs. In this regard, we give a partial solution to an open conjecture: all SRGs satisfy the curvature condition CD(∞, 2). Finally we transition to consider a facility location problem motivated by using Unmanned Aerial Vehicles (UAVs) to guard a border. Here, we find a greedy algorithm, acting on local geometric information, which finds a near optimal placement of base stations for the guarding of UAVs