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Accurate computation of the medial axis of a polyhedron

Authors:

Tim Culver,

John Keyser,

Dinesh ManochaAuthors Info & Claims

SMA '99: Proceedings of the fifth ACM symposium on Solid modeling and applications

Pages 179 - 190

https://doi.org/10.1145/304012.304030

Published: 01 June 1999 Publication History

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References

[1]

S.S. Abhyankar and C. Bajaj. Automatic parametrizatiol.as of rational curves and surfaces iii: Algebraic plane curves. Computer Aided Geometric Design, 5:309-321, 1988.

Digital Library

Google Scholar

[2]

P. Agarwal, B. Aronov, and M. Sharir. Computing envelopes in four dimensions with applications. SlAM J. Comput., 26:1714--1732, 1997.

Digital Library

Google Scholar

[3]

S. Arnborg and H. Feng. Algebraic decomposition of regular curves. J. Symbolic Comput., 15(1):131-140, 1988.

Digital Library

Google Scholar

[4]

M. O. Benouamer, D. Michelucci, and B. Peroche. Error-free boundary evaluation based on a lazy rational arithmetic: A detailed implementation. Computer Aided Design, 26(6):403- 416, June 1994.

Crossref

Google Scholar

[5]

H. Blum. A transformation for extracting new descriptors of shape. In W. Wathen-Durm, editor, Models for the Perception of Speech and Visual Form, pages 362-380. MIT Press, 1'967.

Google Scholar

[6]

J. W. Brandt. Describing a solid with the three-dimensional skeleton. In J. D. Warren, editor, Proceedings of the International Society for Optical Engineering Volume 1830, Curves and Surfaces in Computer Vision and Graphics 111, pages 258-269. SPIE, Boston, 1992.

Google Scholar

[7]

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

Digital Library

Google Scholar

[8]

H. Bronnimann and M. Yvinec. Efficient exact evaluation of signs of determinants. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 166-173, 1997.

Digital Library

Google Scholar

[9]

J.S. Brown and J.F. Traub. On Euclid's algorithm and the theory of subresultant. Journal ofA CM, 18(4):505-514, 1971.

Digital Library

Google Scholar

[10]

J.E Canny. The Complexity of Robot Motion Planning. ACM Doctoral Dissertation Award. MIT Press, 1988.

Digital Library

Google Scholar

[11]

C.-S. Chiang. The Euclidean distance transform. Ph.D. thesis, Dept. Comput. Sci., Purdue Univ., West Lafayette, IN, August 1992. Report CSD-TR 92-050.

Digital Library

Google Scholar

[12]

K.L. Clarkson. Safe and effective determinant evaluation. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 387-395, 1992.

Digital Library

Google Scholar

[13]

Tim Culver. Accurate computation of the medial axis of a polyhedron using exact arithmetic. Technical Report TR99- 012, University of North Carolina-Chapel Hill, 1999.

Digital Library

Google Scholar

[14]

J. H. Davenport. Computer Algebra Systems and algorithms for algebraic computation. Academic Press, London, 2 edition, 1993.

Digital Library

Google Scholar

[15]

J. Demmel. LAPACK: A portable linear algebra library for supercomputers. In Proceedings of the 1989 IEEE Control Systems Society Workshop on Computer-Aided Control System Design, Tampa, FL, Dec 1989. IEEE.

Crossref

Google Scholar

[16]

A. L. Dixon. The eliminant of the equations of four quadric surfaces. Proc. London Math. Soc., pages 340-352, 1910.

Crossref

Google Scholar

[17]

D. Dutta and C. M. Hoffmann. A geometric investigation of the skeleton of CSG objects. In Proc. ASME Conf. Design Automation, 1990.

Crossref

Google Scholar

[18]

D. Dutta and C. M. Hoffmann. On the skeleton of simple CSG objects. Journal of Mechanical Design, ASME Transactions, 115(I):87-94, 1993.

Google Scholar

[19]

G. Elber and M-S. Kim. The bisector surface of freeform rational space curves. Technical Report CIS Report #9619, Technion-Israel institute of Technology, September 1996.

Google Scholar

[20]

S. Fortune. Polyhedral modeling with exact arithmetic. Proceedings of ACM Solid Modeling, pages 225-234, 1995.

Digital Library

Google Scholar

[21]

S. J. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987.

Digital Library

Google Scholar

[22]

LiDIA Group. A library for computational number theory. Technical report, TH Darmstadt, Fachbereich Informatik, Institut fur Theoretische Informatik, 1997.

Google Scholar

[23]

M. Held. On computing Voronoi diagrams of convex polyhedra by means of wavefront propagation. In Proc. 6th Canad. Conf. Comput. Geom., pages 128-133, 1994.

Google Scholar

[24]

Martin Held. Voronoi diagrams and offset curves of curvilinear polygons. Computer-Aided Design, 30(4):287-300, 1998.

Crossref

Google Scholar

[25]

C. M. Hoffmann. How to construct the skeleton of csg objects, in A. Bowyer and J. Davenport, editors, Proceedings of the Fourth IMA Conference, The Mathematics of Surfaces, University of Bath, UK, September 1990. Oxford University Press, New York, ~994.

Google Scholar

[26]

C.M. Hoffmann. Geometric and Solid Modeling. Morgan Kaufmann, San Mateo, California, 1989.

Digital Library

Google Scholar

[27]

Kenneth E. Hoff Iii, Tim Culver, John Keyser, Ming Lin, and Dinesh Manocha. Fast computation of generalized voronoi diagrams using graphics hardware. Technical Report TR99- 011, University of North Carolina-Chapel Hill, 1999.

Digital Library

Google Scholar

[28]

T. Imai. A topology oriented algorithm for the Voronoi diagram of polygons. In Proc. 8th Canad. Conf. Comput. Geom., pages 107-112. C;u'leton University Press, Ottaw~ Canada, 1996.

Digital Library

Google Scholar

[29]

J.K. Johnstone. The Sorting of points along an algebraic curve. Phi) thesis, Comell University, Departmenl: of Computer Science, 1987.

Digital Library

Google Scholar

[30]

J. Keyser, S. Krishnan, and D. Manocha. Efficient and accurate b-rep generation of low degree sculptured solids using exact arithmetic. }In ACM/SIGGRAPH Symposium on Solid Modeling, pages 42-55, 1997.

Digital Library

Google Scholar

[31]

john Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. Mapc: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annu. ACM Sympos. Comput. Geom., 1999.

Digital Library

Google Scholar

[32]

Shankar Krishnan. Efficient and Accurate Boundary Evaluation Algorithms .for Boolean Combinations of Sculptured Solids. PhD thesis, University of North Carolina-Chapel Hill, 1997.

Digital Library

Google Scholar

[33]

D.T. Lee. Medial axis transformation of a planar shape. IEEE Trans. Pattern Anal. Mach. Intell., PAMI-&363-369, 1982.

Google Scholar

[34]

ES. Macaulay. On .some formula in elimination. Proceedings of London Mathematical Society, 1 (33):3-27, May tt902.

Google Scholar

[35]

V. Milenkovic. Robust construction of the Voronoi diagram of a polyhedron. In Proc. 5th Canad. Conf. Comput. Geom., pages 473-478, 1993.

Google Scholar

[36]

P. S. Milne. On the solutions of a set of polynomial equations. In Symbolic and Numerical Computation for Artificial Intelligence, pages 89-102, 1992.

Google Scholar

[37]

P. Pedersen. Multivariate sturm theory. In Proceedings of AAECC, pages 318-332. Springer-Verlag, 1991.

Digital Library

Google Scholar

[38]

Jayachandra Reddy and George Turkiyyah. Comp~tation of 3D skeletons using a generalized Delaunay triangulation technique. Computer-Aided Design, 27(9):677-694, 1995.

Google Scholar

[39]

R. Seidel. Linear programming and convex hulls made easy. In Proc. 6th Ann. A CM Conf. on Computational Geometry, pages 211-215, Berkeley, California, 1990.

Digital Library

Google Scholar

[40]

D. J. Sheehy, C. G. Armstrong, and D. J. Robinson. Computing the medial surface of a solid from a domain Delaunay triangulation. In Proc. A CM/IEEE Syrup. on Solid Modeling and Applications, may 1995.

Digital Library

Google Scholar

[41]

Evan C. Sherbrooke, Nicholas M. Patrikalakis, and Erik Brisson. Computation of the medial axis transform of 3-D polyhedra. In Solid Modeling, pages 187-199. ACM, 1995.

Digital Library

Google Scholar

[42]

V. Srinivasan, L. R. Nackman, J.-M. Tang, and S. N. Meshkat. Automatic mesh generation using the symmetric axis transform of polygonal domains. Proc. IEEE, 80(9):1485-1501, September 1992.

Crossref

Google Scholar

[43]

K. Sugihara and M. Iri. A solid modeling system free from topological inconsistencies. J. Inf. Proc., Inf. Proc. Soc. q( Japan, 12(4):380-393, 1989.

Digital Library

Google Scholar

[44]

E J. Vermeer. Medial Axis Transform to Boundary Represen.- tation Conversion. PhD thesis, CS Dept., Purdue University, West Lafayette, Indiana 47907-1398, USA, 1994.

Digital Library

Google Scholar

[45]

Jules Vleugels and Mark Overmars. Approximating general-. ized Voronoi diagrams in any dimension. Technical R epol~: UU-CS- 1995-14, Department of Computer Science, Utrecht University, 1995.

Google Scholar

[46]

E E. Wolter. Cut locus and medial axis in global shape interrogation and representation. Computer Aided Geometric" Design, 1992.

Google Scholar

[47]

J. Yu. Exact arithmetic solid modeling. PhD thesis, Purdue. University, 1992.

Digital Library

Google Scholar

Cited By

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Conchuir BGardner KJordan KBray DAnderson RJohnston MSwope WHarrison ASheehy DPeters T(2020)Efficient Algorithm for the Topological Characterization of Worm-like and Branched Micelle Structures from SimulationsJournal of Chemical Theory and Computation10.1021/acs.jctc.0c0031116:7(4588-4598)Online publication date: 16-Jun-2020
https://doi.org/10.1021/acs.jctc.0c00311
Attali DNguyen TSivignon I(2019)($$\delta ,{\varepsilon }$$?,?)-Ball Approximation of a ShapeDiscrete & Computational Geometry10.1007/s00454-018-0019-861:3(595-625)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s00454-018-0019-8
Zhang XXia YWang JYang ZTu CWang W(2015)Medial axis tree-an internal supporting structure for 3D printingComputer Aided Geometric Design10.1016/j.cagd.2015.03.01235:C(149-162)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.cagd.2015.03.012
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Index Terms

Accurate computation of the medial axis of a polyhedron
1. Applied computing
  1. Physical sciences and engineering
    1. Engineering
2. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models

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We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding ...

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Published In

SMA '99: Proceedings of the fifth ACM symposium on Solid modeling and applications

June 1999

327 pages

ISBN:1581130805

DOI:10.1145/304012

Chairmen:
Deba Dutta
Univ. of Michigan, Ann Arbor
,
Willem F. Bronsvoort
Delft Univ. of Technology, Delft, The Netherlands
,
David C. Anderson
Purdue Univ., Lafayette, IN

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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New York, NY, United States

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Published: 01 June 1999

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SM99: 5th Symposium on Solid Modeling and Applications, 1999

June 8 - 11, 1999

Michigan, Ann Arbor, USA

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Conchuir BGardner KJordan KBray DAnderson RJohnston MSwope WHarrison ASheehy DPeters T(2020)Efficient Algorithm for the Topological Characterization of Worm-like and Branched Micelle Structures from SimulationsJournal of Chemical Theory and Computation10.1021/acs.jctc.0c0031116:7(4588-4598)Online publication date: 16-Jun-2020
https://doi.org/10.1021/acs.jctc.0c00311
Attali DNguyen TSivignon I(2019)($$\delta ,{\varepsilon }$$?,?)-Ball Approximation of a ShapeDiscrete & Computational Geometry10.1007/s00454-018-0019-861:3(595-625)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s00454-018-0019-8
Zhang XXia YWang JYang ZTu CWang W(2015)Medial axis tree-an internal supporting structure for 3D printingComputer Aided Geometric Design10.1016/j.cagd.2015.03.01235:C(149-162)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.cagd.2015.03.012
Zhao XWang HKomura T(2014)Indexing 3D Scenes Using the Interaction Bisector SurfaceACM Transactions on Graphics10.1145/257486033:3(1-14)Online publication date: 2-Jun-2014
https://dl.acm.org/doi/10.1145/2574860
Huang QGuibas LMitra N(2014)Near-Regular Structure Discovery Using Linear ProgrammingACM Transactions on Graphics10.1145/253559633:3(1-17)Online publication date: 2-Jun-2014
https://dl.acm.org/doi/10.1145/2535596
Piegl LKulkarni PRajab K(2014)Data Processing for Medial Axis Computation Using B-Spline SmoothingJournal of Computing and Information Science in Engineering10.1115/1.402799114:4(041002)Online publication date: 1-Sep-2014
https://doi.org/10.1115/1.4027991
Xia SDing NJin MWu HYang Y(2013)Medial axis construction and applications in 3D wireless sensor networks2013 Proceedings IEEE INFOCOM10.1109/INFCOM.2013.6566784(305-309)Online publication date: Apr-2013
https://doi.org/10.1109/INFCOM.2013.6566784
Han LHu JLi L(2013)Structure Descriptor for Articulated Shape AnalysisProceedings, Part I, of the 9th International Symposium on Advances in Visual Computing - Volume 803310.1007/978-3-642-41914-0_18(171-180)Online publication date: 29-Jul-2013
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Bhatt AKhurieshi MSiddhartha (2012)Validation of Medial Axis Transform ObjectsComputer-Aided Design and Applications10.3722/cadaps.2012.517-5299:4(517-529)Online publication date: Jan-2012
https://doi.org/10.3722/cadaps.2012.517-529
Ba WCao LLiu J(2012)Research on 3D medial axis transform via the saddle point programming methodComputer-Aided Design10.1016/j.cad.2012.07.00144:12(1161-1172)Online publication date: 1-Dec-2012
https://dl.acm.org/doi/10.1016/j.cad.2012.07.001
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Computing the medial axis of a polyhedron reliably and efficiently

Medial-axis-driven shape deformation with volume preservation

Exact computation of the medial axis of a polyhedron

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Computing the medial axis of a polyhedron reliably and efficiently

Medial-axis-driven shape deformation with volume preservation

Exact computation of the medial axis of a polyhedron

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