ABSTRACT
We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance µ in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution shape P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. An alternative algorithm, based purely on rational arithmetic, answers the same deconstruction problem, up to an uncertainty parameter, and its running time depends on the parameter δ (in addition to the other input parameters: n, δ and the radius of the disk). If the input shape is found to be approximable, the rational-arithmetic algorithm also computes an approximate solution shape for the problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one. Our study is motivated by applications from two different domains. However, since the offset operation has numerous uses, we anticipate that the reverse question that we study here will be still more broadly applicable. We present results obtained with our implementation of the rational-arithmetic algorithm.
- Hansen, A., Arbab, F.: An algorithm for generating NC tool paths for arbitrarily shaped pockets with islands. ACM Trans. Graph. 11 (April 1992) 152--182 Google ScholarDigital Library
- Matheron, G.: Random sets and integral geometry. Wiley New York, (1974)Google Scholar
- Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Inc., Orlando, FL, USA (1983) Google ScholarDigital Library
- Damen, J., van Kreveld, M., Spaan, B.: High quality building generalization by extending the morphological operators. In: 11th ICA Workshop on Generalisation and Multiple Reslides. (June 2008)Google Scholar
- Drysdale, R., Rote, G., Sturm, A.: Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs. Computational Geometry, Theory and Applications 41(1--2) (2008) 31 -- 47 Google ScholarDigital Library
- Heimlich, M., Held, M.: Biarc approximation, simplification and smoothing of polygonal curves by means of Voronoi-based tolerance bands. Int. J. Comput. Geometry Appl. 18(3) (2008) 221--250Google ScholarCross Ref
- Sallee, G.T.: Minkowski decomposition of convex sets. Israel Journal of Mathematics 12(3) (September 1982) 266--276Google ScholarCross Ref
- Ostrowski, A.: Über die Bedeutung der Theorie der konvexen Polyeder für die formale Algebra. Jahresberichte Deutsche Math. Ver. 30 (1921) 98--99 English version: "On the Significance of the Theory of Convex Polyhedra for Formal Algebra" ACM Sigsam Bulletin 33 (1999). Google ScholarDigital Library
- Gao, S., Lauder, A.: Decomposition of polytopes and polynomials. Discrete & Computational Geometry 26 (2001) 89--104Google ScholarDigital Library
- Emiris, I., Tsigaridas, E.: Minkowski decomposition of convex lattice polygons. In: Algebraic geometry and geometric modeling. Springer (2006) 217--236Google ScholarCross Ref
- Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. 3rd edn. Springer-Verlag (October 1990)Google Scholar
- Aggarwal, A., Booth, H., O'Rourke, J., Suri, S., Yap, C.K.: Finding minimal convex nested polygons. Inf. Comput. 83(1) (1989) 98--110 Google ScholarDigital Library
- Berberich, E., Halperin, D., Kerber, M., Pogalnikova, R.: Polygonal reconstruction from approximate offsets. In Vahrenhold, J., ed.: 26th European Workshop on Computational Geometry: Workshop Proceedings, Dortmund, Germany, TU Dortmund (March 2010) 65--68Google Scholar
- Yap, C.K.: An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete & Computational Geometry 2 (1987) 365--393Google ScholarDigital Library
- Held, M.: On the Computational Geometry of Pocket Machining. Volume 500 of Lecture Notes in Computer Science. Springer (1991) Google ScholarDigital Library
- Canny, J., Donald, B., Ressler, E.K.: A rational rotation method for robust geometric algorithms. In: SCG '92: Proceedings of the Eighth Annual Symposium on Computational geometry, New York, NY, USA, ACM (1992) 251--260 Google ScholarDigital Library
- Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry 1(1) (1986) 59--71 Google ScholarDigital Library
- Leven, D., Sharir, M.: Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams. Discrete & Computational Geometry 2 (1987) 9--31Google ScholarDigital Library
- Andrew, A.M.: Another efficient algorithm for convex hulls in two dimensions. Inform. Process. Lett. 9(5) (1979) 216--219Google ScholarCross Ref
- Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Inform. Process. Lett. 1 (1972) 132--133Google ScholarCross Ref
- Giezeman, G.J., Wesselink, W.: 2D polygons. In: CGAL User and Reference Manual. 3.7 edn. CGAL Editorial Board (2010) http://www.cgal.org/Manual/3.7/doc_html/cgal_manual/packages.html#Pkg:P%olygon2.Google Scholar
- Wein, R.: Exact and approximate construction of offset polygons. Computer-Aided Design 39(6) (2007) 518--527 Google ScholarDigital Library
- Wein, R.: 2D Minkowski sums. In: CGAL User and Reference Manual. 3.7 edn. CGAL Editorial Board (2010) http://www.cgal.org/Manual/3.7/doc_html/cgal_manual/packages.html#Pkg:M%inkowskiSum2.Google Scholar
- Fogel, E., Wein, R., Zukerman, B., Halperin, D.: 2D regularized Boolean set-operations. In: CGAL User and Reference Manual. 3.7 edn. CGAL Editorial Board (2010) http://www.cgal.org/Manual/3.7/doc_html/cgal_manual/packages.html#Pkg:B%ooleanSetOperations2.Google Scholar
Index Terms
- Deconstructing approximate offsets
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