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Degeneracy and Instability in Geometric Computation [chapter]

Kokichi Sugihara
2001 Geometric Modelling  
Instability in geometric computation comes from numerical errors, which first cause misjudgements of geometric structure, and then generate topological inconsistency.  ...  This paper describes main sources of instability in geometric computation, and reviews two mutually opposite approaches to this problem.  ...  The symbolic perturbation scheme enables us to ignore tedious exceptional treatment of degenerate cases, but we have to be careful in the use of the output for the degenerate input.  ... 
doi:10.1007/978-0-387-35490-3_1 fatcat:mllkwvn7ibf7rkptzgrje5t4ha

Page 560 of Current Opinion Vol. 35, Issue 5 [page]

1903 Current Opinion  
In speaking of the process of conventional- ization or degeneration of realistic motives, Profwssor Hamlin says: ‘‘ Indeed, this degener- ation may reasonably be accepted as suggesting that the geometric  ...  forms which it approaches were already in habitual use when it began, and that the direction of the degeneration was determined by a pre-existing habit or ‘ex- pectancy’ of geometric form acquired in  ... 

Geometry of the transport equation in multicomponent WKB approximations

C. Emmrich, A. Weinstein
1996 Communications in Mathematical Physics  
In this paper we give a completely geometric derivation of the transport equation, without using local sections and without assuming complete diagonalizability of the matrix valued principal symbol, or  ...  In the non-degenerate case this term may be rewritten in an even simpler geometric form.  ...  The research of both authors was partially supported by DOE Contract DE-FG03-93ER25177.  ... 
doi:10.1007/bf02099256 fatcat:vorjxx6csbeedmvejcjbengn7i

Automatic deduction in (dynamic) geometry: Loci computation

Francisco Botana, Miguel A. Abánades
2014 Computational geometry  
The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial  ...  Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry.  ...  Finally, we want to acknowledge the work and suggestions of the referees, that have contributed to substantially improve our work.  ... 
doi:10.1016/j.comgeo.2013.07.001 fatcat:icnlv3eouzazbpsl2k5o6sgygu

A deterministic pseudorandom perturbation scheme for arbitrary polynomial predicates [article]

Geoffrey Irving, Forrest Green
2013 arXiv   pre-print
We present a symbolic perturbation scheme for arbitrary polynomial geometric predicates which combines the benefits of Emiris and Canny's simple randomized linear perturbation scheme with Yap's multiple  ...  For nonmaliciously chosen predicates, our method is as fast as the linear scheme, scaling reasonably with the degree of the polynomial even for fully degenerate input.  ...  We describe symbolic perturbation in the framework of nonstandard analysis; see [19] , [8] , and [17] for the geometric meaning of this approach.  ... 
arXiv:1308.1986v1 fatcat:hn6bhmmruzak5dccqjef4z4cgu

Using Three-Valued Logic to Specify and Verify Algorithms of Computational Geometry [chapter]

Jens Brandt, Klaus Schneider
2005 Lecture Notes in Computer Science  
Therefore, we use a three-valued logic for the definition of geometric primitives to explicitly handle such degenerate cases.  ...  a geometric primitive.  ...  Explicit Treatment of Degeneracies. The explicit treatment of degeneracies suffers from the enormous number of cases.  ... 
doi:10.1007/11576280_28 fatcat:4se43sbd3rdghbtb7jt6xfppde

Handling degenerate cases in exact geodesic computation on triangle meshes

Yong-Jin Liu, Qian-Yi Zhou, Shi-Min Hu
2007 The Visual Computer  
By applying these geometric algorithms to real-world data, degenerate cases frequently appear. In this paper we classify and enumerate all the degenerate cases in a systematic way.  ...  The computation of exact geodesics on triangle meshes is a widely used operation in computeraided design and computer graphics.  ...  In geometric computation, degenerate cases will increase the instability of the generic algorithm. Theoretically, degenerate cases can be handled by using the symbolic perturbation scheme [1] .  ... 
doi:10.1007/s00371-007-0136-5 fatcat:fctgzdylybe2nlegvl3gv2kls4

Asymptotics of relativistic spin networks

John W Barrett, Christopher M Steele
2003 Classical and quantum gravity  
For the 4-simplex (10j-symbol) the asymptotic formula is compared with numerical calculations of the Spin Network evaluation. Finally we discuss the asymptotics of the SO(3,1) 10j-symbol.  ...  For the tetrahedral spin network this gives the square of the Ponzano-Regge asymptotic formula for the SU(2) 6j symbol.  ...  An interesting feature of our analysis, carried out using the stationary phase approximation to an integral, is that it has both 'degenerate' and 'non-degenerate' geometrical configurations.  ... 
doi:10.1088/0264-9381/20/7/307 fatcat:vmhqwabfrjbgvp3tw3t4vo25qi

Sketch-based pruning of a solution space within a formal geometric constraint solver

C. Essert-Villard, P. Schreck, J.-F. Dufourd
2000 Artificial Intelligence  
In CAD systems, formal geometric solvers enable the designer to draw a sketch and to provide constraints that are compiled into a construction plan by symbolic geometric reasoning.  ...  But when the number of solutions becomes too high, it is very time-consuming to examine each of them to determine which one is the closest to the user's will.  ...  Since the symbolic resolution of a system S consists in building a triangular solved system T (see [9] ) which is equivalent to S, the degenerate cases of S are the degenerate cases of T , and conversely  ... 
doi:10.1016/s0004-3702(00)00061-8 fatcat:pisfezbuond7fnxt2plalqizly

Classifying Topoi and Preservation of Higher Order Logic by Geometric Morphisms [article]

Shawn J. Henry
2013 arXiv   pre-print
We set up a deduction calculus for internal geometric theories and show that it proves a contradiction if and only if the classifying topos of that theory is degenerate.  ...  order theory which says "the classifying topos for T is degenerate".  ...  ACKNOWLEDGEMENTS I am most indebted to my advisor, Andreas Blass, for his patience and for sharing a small part of his immense knowledge with me.  ... 
arXiv:1305.3254v1 fatcat:345spnufbfadfb6d4dnk4owsya

Hexahedral Mesh Generation using the Embedded Voronoi Graph

A. Sheffer, M. Etzion, A. Rappoport, M. Bercovier
1999 Engineering with Computers  
The embedded Voronoi graph contains the full symbolic information of the Voronoi diagram and the medial axis of the object, and a geometric approximation to the real geometry.  ...  It handles any volume, even if its medial axis is degenerate.  ...  Part of this research was supported by the Israeli Ministry of Science, the program for development of scientific and technological infrastructures.  ... 
doi:10.1007/s003660050020 fatcat:2bwxoifa5zccvenhmok5t4huaa

Qualitative Symbolic Perturbation

Olivier Devillers, Menelaos Karavelas, Monique Teillaud, Marc Herbstritt
2016 International Symposium on Computational Geometry  
In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in ε for the inputs of a predicate.  ...  Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.  ...  Introduction In earlier computational geometry papers, the treatment of degenerate configurations was mainly ignored. However, degenerate situations actually do occur in practice.  ... 
doi:10.4230/lipics.socg.2016.33 dblp:conf/compgeom/DevillersKT16 fatcat:lonebrclfbai3mttgprqgyewte

Qualitative symbolic perturbation: two applications of a new geometry-based perturbation framework

Olivier Devillers, Menelaos Karavelas, Monique Teillaud
2015 Journal of Computational Geometry  
In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in $\varepsilon$ for the inputs of a predicate.  ...  Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.  ...  Acknowledgments The authors warmly thank the reviewers for their very careful reading of the paper and their detailed comments and suggestions.  ... 
doi:10.20382/jocg.v8i1a11 dblp:journals/jocg/DevillersKT17 fatcat:2cyalkxqdramzg6gbtrqufocue

Bispectrum covariance in the flat-sky limit

B. Joachimi, X. Shi, P. Schneider
2009 Astronomy and Astrophysics  
Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol,  ...  We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding.  ...  B.J. acknowledges support by the Deutsche Telekom Stiftung and the Bonn-Cologne Graduate School of Physics and Astronomy.  ... 
doi:10.1051/0004-6361/200912906 fatcat:uusbxyfpezfkpp5q7hers64tnm

Page 1301 of Mathematical Reviews Vol. 56, Issue 4 [page]

1978 Mathematical Reviews  
This enables one to define the symbol of A as an object on A, recovered from geometric information on the asymptotic behavior of <e”u, A> for various %, once the geometry has been studied in Chapter 3.  ...  In particular, it should be pointed out that the brief treatment of L? continuity in Theorem 2.5.4 for pseudodifferential operators with symbols in So ,_, is misleading. L?  ... 
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