Robustness and Precision Issues in Geometric Computation

This is a preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. We give a survey on techniques that have been proposed and successfully used to attack robustness and precision problems in the implementation of geometric algorithms. Work on this survey was partially supported by the ESPRIT IV Long Term Research Project No. 21957 (CGAL).

[1]  Douglas M. Priest.  On properties of floating point arithmetics: numerical stability and the cost of accurate computations , 1992 .

[2]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[3]  Geert-Jan Giezeman,et al.  The CGAL Kernel: A Basis for Geometric Computation , 1996, WACG.

[4]  Chandrajit L. Bajaj,et al.  Convex Decomposition of Polyhedra and Robustness , 1992, SIAM J. Comput..

[5]  John E. Hopcroft,et al.  Towards implementing robust geometric computations , 1988, SCG '88.

[6]  Shiaofen Fang,et al.  Robustness in Geometric Modeling - Tolerance-Based Methods , 1991, Workshop on Computational Geometry.

[7]  Lee R. Nackman,et al.  Finding compact coordinate representations for polygons and polyhedra , 1990 .

[8]  Rolf Klein,et al.  Algorithmische Geometrie , 1997 .

[9]  Stefan Näher.  LEDA, a Platform for Combinatorial and Geometric Computing , 1997, Handbook of Data Structures and Applications.

[10]  Kurt Mehlhorn,et al.  On degeneracy in geometric computations , 1994, SODA '94.

[11]  Mark Segal,et al.  Using tolerances to guarantee valid polyhedral modeling results , 1990, SIGGRAPH.

[12]  A. James Stewart.  Local Robustness and its Application to Polyhedral Intersection , 1994, Int. J. Comput. Geom. Appl..

[13]  Mariette Yvinec,et al.  Algorithmic geometry , 1998 .

[14]  Giuseppe Liotta,et al.  Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design , 1998, SIAM J. Comput..

[15]  Kokichi Sugihara,et al.  A Robust and Consistent Algorithm for Intersecting Convex Polyhedra , 1994, Comput. Graph. Forum.

[16]  Jerzy W. Jaromczyk,et al.  Computing Convex Hull in a Floating Point Arithmetic , 1994, Comput. Geom..

[17]  Kokichi Sugihara,et al.  Delaunay triangulations in three dimensions with finite precision arithmetic , 1992, Comput. Aided Geom. Des..

[18]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[19]  K. Mehlhorn,et al.  The LEDA class real number , 1996 .

[20]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[21]  Carlo H. Séquin,et al.  Partitioning polyhedral objects into nonintersecting parts , 1988, IEEE Computer Graphics and Applications.

[22]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[23]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[24]  Peter Schorn,et al.  Degeneracy in Geometric Computation and the Perturbation Approach , 1994, Comput. J..

[25]  Jonathan Richard Shewchuk,et al.  Robust adaptive floating-point geometric predicates , 1996, SCG '96.

[26]  Christoph M. Hoffmann,et al.  The problems of accuracy and robustness in geometric computation , 1989, Computer.

[27]  A. R. Forrest.  Computational Geometry in Practice , 1991 .

[28]  Kokichi Sugihara,et al.  Numerically Robust Algorithm for Contructing Constrained Delaunay Triangulation , 1994, CCCG.

[29]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[30]  Victor J. Milenkovic,et al.  Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic , 1989, 30th Annual Symposium on Foundations of Computer Science.

[31]  Kurt Mehlhorn,et al.  LEDA: a platform for combinatorial and geometric computing , 1997, CACM.

[32]  Steven Fortune,et al.  Robustness Issues in Geometric Algorithms , 1996, WACG.

[33]  Ketan Mulmuley,et al.  Computational geometry - an introduction through randomized algorithms , 1993 .

[34]  Jonathan Richard Shewchuk,et al.  Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates , 1997, Discret. Comput. Geom..

[35]  Giuseppe Liotta,et al.  Robust proximity queries: an illustration of degree-driven algorithm design , 1997, SCG '97.

[36]  Victor J. Milenkovic,et al.  Numerical stability of algorithms for line arrangements , 1991, SCG '91.

[37]  Victor Y. Pan,et al.  Computing exact geometric predicates using modular arithmetic with single precision , 1997, SCG '97.

[38]  Christopher J. Van Wyk,et al.  Efficient exact arithmetic for computational geometry , 1993, SCG '93.

[39]  Raimund Seidel,et al.  Efficient Perturbations for Handling Geometric Degeneracies , 1997, Algorithmica.

[40]  Jürg Nievergelt,et al.  Das Rätsel der verzopften Geraden - Overflow , 1988, Inform. Spektrum.

[41]  J. Blomer.  Computing sums of radicals in polynomial time , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[42]  David P. Dobkin,et al.  Applied Computational Geometry: Towards Robust Solutions of Basic Problems , 1990, J. Comput. Syst. Sci..

[43]  Kurt Mehlhorn,et al.  The Implementation of Geometric Algorithms , 1994, IFIP Congress.

[44]  F. Frances Yao,et al.  Finite-resolution computational geometry , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[45]  Maurice Mignotte,et al.  Identification of Algebraic Numbers , 1982, J. Algorithms.

[46]  Dinesh Manocha,et al.  Applied Computational Geometry Towards Geometric Engineering , 1996, Lecture Notes in Computer Science.

[47]  Chee-Keng Yap,et al.  A geometric consistency theorem for a symbolic perturbation scheme , 1988, SCG '88.

[48]  D. Pullar.  CONSEQUENCES OF USINGA TOLERANCE PARADIGM IN SPATIAL OVERLAY , 2022 .

[49]  Zhenyu Li,et al.  Constructing strongly convex hulls using exact or rounded arithmetic , 1990, SCG '90.

[50]  V. Milenkovic.  Robust Polygon Modeling , 1993 .

[51]  Mariette Yvinec,et al.  Efficient Exact Evaluation of Signs of Determinants , 1997, Symposium on Computational Geometry.

[52]  Kouji Ouchi.  Real/Expr: Implementation of an Exact Computation Package , 1997 .

[53]  Victor J. Milenkovic,et al.  Robust polygon modelling , 1993, Comput. Aided Des..

[54]  Franco P. Preparata,et al.  A Probabilistic Analysis of the Power of Arithmetic Filters , 1998, Discret. Comput. Geom..

[55]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[56]  Michael F. Goodchild,et al.  Issues of quality and uncertainty , 1991 .

[57]  Franco P. Preparata,et al.  Computational Geometry , 1985, Texts and Monographs in Computer Science.

[58]  Sudhir P. Mudur,et al.  Interval Methods for Processing Geometric Objects , 1984, IEEE Computer Graphics and Applications.

[59]  Peter Schorn.  An Axiomatic Approach to Robust Geometric Programs , 1993, J. Symb. Comput..

[60]  A. James Stewart.  Robust Point Location in Approximate Polygons , 1991 .

[61]  Wei Chen,et al.  Parallel robust algorithms for constructing strongly convex hulls , 1996, SCG '96.

[62]  M. Karasick.  On the representation and manipulation of rigid solids , 1989 .

[63]  Kokichi Sugihara,et al.  On Finite-Precision Representations of Geometric Objects , 1989, J. Comput. Syst. Sci..

[64]  Leonidas J. Guibas,et al.  Epsilon geometry: building robust algorithms from imprecise computations , 1989, SCG '89.

[65]  Philippe Jaillon,et al.  A "Lazy" Solution to Imprecision in Computational Geometry , 1993 .

[66]  Kenneth L. Clarkson,et al.  Safe and effective determinant evaluation , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[67]  Peter Schorn.  Robust algorithms in a program library for geometric computation , 1991, Informatik-Dissertationen ETH Zürich.

[68]  Kurt Mehlhorn,et al.  How to Compute the Voronoi Diagram of Line Segments: Theoretical and Experimental Results , 1994, ESA.

[69]  A. James Stewart.  The Theory and Pratice of Robust Geometric Computation, or, How toBuild Robust Solid Modelers , 1991 .

[70]  M. Laszlo.  Computational Geometry and Computer Graphics in C , 1995 .

[71]  T. J. Dekker,et al.  A floating-point technique for extending the available precision , 1971 .

[72]  Giorgio Gambosi,et al.  A convex hull algorithm for points with approximately known positions , 1994, Int. J. Comput. Geom. Appl..

[73]  Jean-Daniel Boissonnat,et al.  Robust Plane Sweep for Intersecting Segments , 2000, SIAM J. Comput..

[74]  Dirk Riehle,et al.  Domain-driven framework layering in large systems , 2000, CSUR.

[75]  Chee-Keng Yap,et al.  Towards Exact Geometric Computation , 1997, Comput. Geom..

[76]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[77]  Chee Yap,et al.  The exact computation paradigm , 1995 .

[78]  Douglas M. Priest,et al.  Algorithms for arbitrary precision floating point arithmetic , 1991, [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic.

[79]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[80]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[81]  John F. Canny,et al.  An efficient approach to removing geometric degeneracies , 1992, SCG '92.

[82]  Jean Vuillemin,et al.  BigNum: A Portable and Efficient Package for Arbitrary-Precision Arithmetic , 1989 .

[83]  William L. Ditto,et al.  Principles and applications of chaotic systems , 1995, CACM.

[84]  Steven Fortune,et al.  Stable maintenance of point set triangulations in two dimensions , 1989, 30th Annual Symposium on Foundations of Computer Science.

[85]  Michael D. Hirsch,et al.  A Robust Algorithm for Point in Polyhedron , 1993, CCCG.

[86]  Chee Yap.  Symbolic treatment of geometric degeneracies: Proceedings of the International IFIPS Conference on System Modeling and Optimization. Tokyo, 1987 , 1987 .

[87]  John F. Canny,et al.  A General Approach to Removing Degeneracies , 1995, SIAM J. Comput..

[88]  Kokichi Sugihara,et al.  A robust Topology-Oriented Incremental algorithm for Voronoi diagrams , 1994, Int. J. Comput. Geom. Appl..

[89]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[90]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry , 2012, EATCS Monographs on Theoretical Computer Science.

[91]  Mark H. Overmars.  Designing the Computational Geometry Algorithms Library CGAL , 1996, WACG.

[92]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[93]  Klaus H. Hinrichs,et al.  Algorithms and data structures - with applications to graphics and geometry , 1999, vdf Lehrbuch.

[94]  Kokichi Sugihara,et al.  A solid modelling system free from topological inconsistency , 1990 .

[95]  Jack Snoeyink,et al.  On the bit complexity of minimum link paths: superquadratic algorithms for problems solvable in linear time , 1996, SCG '96.

[96]  Kevin G. Suffern,et al.  Interval methods in computer graphics , 1991, Comput. Graph..

[97]  John E. Hopcroft,et al.  Robust set operations on polyhedral solids , 1987, IEEE Computer Graphics and Applications.

[98]  Thomas Ottmann,et al.  Numerical stability of geometric algorithms , 1987, SCG '87.

[99]  Donald E. Knuth,et al.  Axioms and Hulls , 1992, Lecture Notes in Computer Science.

[100]  Pedro Jussieu de Rezende,et al.  GeoLab: An Environment for Development of Algorithms in Computational Geometry , 1993, CCCG.

[101]  Maurice Mignotte,et al.  Mathematics for computer algebra , 1991 .

[102]  Ioannis Z. Emiris,et al.  A Complete Implementation for Computing General Dimensional Convex Hulls , 1998, Int. J. Comput. Geom. Appl..

[103]  Kurt Mehlhorn,et al.  A strong and easily computable separation bound for arithmetic expressions involving square roots , 1997, SODA '97.

[104]  M. Iri,et al.  Construction of the Voronoi diagram for 'one million' generators in single-precision arithmetic , 1992, Proc. IEEE.

[105]  Carlo H. Séquin,et al.  Consistent calculations for solids modeling , 1985, SCG '85.

[106]  Niklaus Wirth,et al.  Pascal User Manual and Report , 1991, Springer New York.

[107]  John F. Canny,et al.  Generalised Characteristic Polynomials , 1990, J. Symb. Comput..

[108]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[109]  C. Barber.  Computational geometry with imprecise data and arithmetic , 1992 .

[110]  Christopher J. Van Wyk,et al.  Static analysis yields efficient exact integer arithmetic for computational geometry , 1996, TOGS.

[111]  Chandrajit L. Bajaj,et al.  Robust Decompositions of Polyhedra , 1989, FSTTCS.

[112]  Kokichi Sugihara,et al.  Topology-Oriented Divide-and-Conquer Algorithm for Voronoi Diagrams , 1995, CVGIP Graph. Model. Image Process..

[113]  Kokichi Sugihara.  Topologically Consistent Algorithms Realted to Convex Polyhedra , 1992, ISAAC.

[114]  Franco P. Preparata.  Robustness in Geometric Algorithms , 1996, WACG.

[115]  David Goldberg,et al.  What every computer scientist should know about floating-point arithmetic , 1991, CSUR.

[116]  Christoph Burnikel,et al.  Exact computation of Voronoi diagrams and line segment intersections , 1996 .

[117]  Kokichi Sugihara.  An intersection algorithm based on Delaunay triangulation , 1992, IEEE Computer Graphics and Applications.

[118]  Leonidas J. Guibas,et al.  Implementing Geometric Algorithms Robustly , 1996, WACG.

[119]  Dan Halperin,et al.  A perturbation scheme for spherical arrangements with application to molecular modeling , 1997, SCG '97.

[120]  Steven Fortune,et al.  Numerical stability of algorithms for 2-d Delaunay triangulations , 1995, Int. J. Comput. Geom. Appl..

[121]  Leonidas J. Guibas,et al.  Snap rounding line segments efficiently in two and three dimensions , 1997, SCG '97.

[122]  Kokichi Sugihara,et al.  Topology Oriented vs. Exact Arithmetic - Experience in Implementing the Three-Dimensional Convex Hull Algorithm , 1997, ISAAC.

[123]  Victor J. Milenkovic,et al.  An Experiment Using LN for Exact Geometric Computations , 1993, CCCG.

[124]  Raimund Seidel.  The Nature and Meaning of Perturbations in Geometric Computing , 1998, Discret. Comput. Geom..

[125]  Leonidas J. Guibas,et al.  Rounding arrangements dynamically , 1995, SCG '95.

[126]  Victor J. Milenkovic,et al.  Verifiable Implementations of Geometric Algorithms Using Finite Precision Arithmetic , 1989, Artif. Intell..

[127]  Lee R. Nackman,et al.  Efficient Delaunay triangulation using rational arithmetic , 1991, TOGS.