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Degeneracy and Instability in Geometric Computation
[chapter]
2001
Geometric Modelling
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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
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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
1996
Communications in Mathematical Physics
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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
2014
Computational geometry
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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]
2013
arXiv
pre-print
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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]
2005
Lecture Notes in Computer Science
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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
2007
The Visual Computer
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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
2003
Classical and quantum gravity
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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
2000
Artificial Intelligence
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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]
2013
arXiv
pre-print
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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
1999
Engineering with Computers
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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
2016
International Symposium on Computational Geometry
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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
2015
Journal of Computational Geometry
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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
2009
Astronomy and Astrophysics
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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
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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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