On the Computational Geometry of Pocket MachiningIn this monograph the author presents a thorough computational geometry approach to handling theoretical and practical problems arising from numerically controlled pocket machining. The approach unifies two scientific disciplines: computational geometry and mechanical engineering. Topics of practical importance that are dealt with include the selection of tool sizes, the determination of tool paths, and the optimization of tool paths. Full details of the algorithms are given from a practical point of view, including information on implementation issues. This practice-minded approach is embedded in a rigorous theoretical framework enabling concise statement of definitions and proof of the correctness and efficiency of the algorithms. In particular, the construction of Voronoi diagrams and their use for offset calculations are investigated in great detail. Based on Voronoi diagrams, a graph-like structure is introduced that serves as a high-level abstraction of the pocket geometry and provides the basis for algorithmically performing shape interrogation and path planning tasks. Finally, the efficiency and robustness of the approach is illustrated with figures showing pocketing examples that have been processed by the author's own implementation. |
Contents
Introduction | 3 |
11 Computational Geometry | 4 |
12 NC Pocket Machining | 5 |
122 Main Types of NC Path Control | 6 |
123 Motivation for Pocket Machining | 7 |
124 Informal Problem Specification | 9 |
13 Analysis of Prior and Related Work | 11 |
132 State of the Art | 12 |
512 Our Approach to Constructing Voronoi Diagrams | 66 |
52 Basic Concepts | 68 |
522 Definition of the Voronoi Diagram | 69 |
523 Basic Facts about Voronoi Diagrams | 71 |
53 Computing VTC0 | 77 |
532 Discussion of the Voronoi Algorithm | 79 |
54 Analysis of the Voronoi Algorithm | 82 |
543 Worstcase Analysis of the Voronoi Algorithm | 84 |
133 Conventional or Set Theoretical Approach | 13 |
134 Artificial Intelligence Approach | 15 |
Survey of Contourparallel Milling | 17 |
21 Introduction | 18 |
213 Restrictions Imposed on the Pocket | 19 |
222 Using Voronoi Diagrams for Offsetting | 21 |
23 Solving Geometrical Problems of Pocketing | 22 |
232 The Concept of Monotonous Areas | 24 |
233 Computing Optimal Cutter Pass Distances | 25 |
234 Generating the Tool Path | 28 |
24 Pocketing Features of GEoPocKET | 30 |
25 Solving Technological Problems of Pocketing | 31 |
251 Computeraided Tool Selection | 32 |
252 Ensuring a Satisfying Surface Quality Control Parameters for Handling Different Cutting Tools | 33 |
253 Further Technological Features | 35 |
26 Concluding Remarks | 36 |
Survey of Directionparallel Milling | 37 |
31 Introduction | 38 |
32 Motivation and Basic Aspects | 39 |
33 Pocketing Features of ZlGPocKET | 41 |
332 User Assistance and Advanced Technological Features | 42 |
341 Constructing the Mesh | 43 |
342 Computing the Tool Path | 44 |
35 Solving Technological Problems of Pocketing | 46 |
352 Handling Negative Islands | 47 |
353 Avoiding to Drill Unnecessary Holes | 48 |
36 Practical Results and Heuristic Analysis | 49 |
362 How Important Is a Suitable Inclination? | 50 |
363 Zigzag Versus Offset Curve Milling | 51 |
372 Open Problems | 52 |
Preliminaries | 53 |
41 Notational Conventions | 54 |
412 Upper Case Script | 55 |
414 Lower Case Roman | 56 |
42 Topology and Geometry Revisited | 57 |
43 Graph Theory Revisited | 58 |
44 Remarks on Algorithms | 59 |
Contourparallel Milling | 61 |
Computing Voronoi Diagrams | 63 |
51 Introduction | 64 |
55 Computing VVB | 86 |
Implementational Issues | 89 |
61 Representation of the Contours | 90 |
62 Representation of the Bisectors | 91 |
622 Storing the Bisector Parameterizations | 94 |
63 Representation of the Voronoi Diagram | 96 |
632 On the Maximal Number of Analytic Bisectors | 98 |
64 Manipulating Bisectors | 100 |
The Concept of Monotonous Areas | 103 |
71 Introducing Monotonous Areas | 104 |
712 Basic Properties of Monotonous Areas | 105 |
72 Determining Monotonous Areas | 108 |
722 Algorithm for Determining Monotonous Areas | 110 |
73 Implementational Issues | 112 |
732 Handling Parallel Lines and Concentric Arcs | 113 |
Generating the Tool Path | 115 |
81 Optimal Pass Distance | 116 |
812 Characterization of an Optimal Pass Distance | 117 |
82 Computing Optimal Offsets | 118 |
822 Algorithm | 120 |
83 Tool Path Assembly | 122 |
832 Correctness | 124 |
Directionparallel Milling | 127 |
Constructing the Mesh | 129 |
91 The Mesh | 130 |
912 Implementational Issues | 131 |
922 Algorithm Complexity | 134 |
932 Algorithm | 136 |
Generating the Tool Path | 139 |
101 Algorithm for Tool Path Generation | 140 |
1012 Correctness | 143 |
102 Handling Negative Islands | 145 |
Examples | 147 |
A1 Contourparallel Milling | 148 |
A2 Directionparallel Milling | 159 |
List of Figures | 165 |
List of Tables | 167 |
169 | |
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Common terms and phrases
analytic bisectors approach b₁ b₂ Blackboard Bold boundary circular arcs complexity Computational Geometry contained contour elements Contour-parallel Milling convex CPU-time cutter pass distances defined Definition denote depicted in Fig Direction-parallel Milling Edge2 endpoints geometric GEOPOCKET graph Handling Negative Islands Hence inclination inner point innermost point intersect island contours Lemma line segment maximal maximal connected component merge process mesh minimal minimum spanning tree monotonous area multiply-connected NC machine objects offset curves optimal pass distances original nodes outgoing edges P₁ parameterization path-connected Persson pocket area pocket contours Pocket Machining Pocketing Example problems Proceedings programming Proof radius rapid feed reflex profile reflex vertices restricted Right_Node scan simple polygon simply-connected strait tangential technological Theorem tool path total number totally milled triangle inequality upper bound VD(B VD(C VD(PL VD(PR Voronoi Algorithm Voronoi diagram Voronoi polygon VP(OL width worst-case ZIGPOCKET zigzag line zigzag path zigzag segments
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