Near-Optimal Parameterization of the Intersection of Quadrics: I. The Generic Algorithm
Identifieur interne : 006813 ( Main/Merge ); précédent : 006812; suivant : 006814Near-Optimal Parameterization of the Intersection of Quadrics: I. The Generic Algorithm
Auteurs : Laurent Dupont [France] ; Daniel Lazard [France] ; Sylvain Lazard [France] ; Sylvain Petitjean [France]Source :
English descriptors
Abstract
We present the first efficient algorithm for computing an exact parametric representation of the intersection of two quadrics in three-dimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection are rational functions whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, or two conics). Furthermore, the parameterization is near-optimal in the sense that the number of square roots appearing in the coefficients of these functions is minimal except in a small number of cases where there may be an extra square root. In addition, the algorithm is practical: a complete, robust and efficient C++ implementation is described in Part IV [12] of this paper. In Part I, we present an algorithm for computing a parameterization of the intersection of two arbitrary quadrics which we prove to be near-optimal in the generic, smooth quartic, case. Parts II and III [4, 5] treat the singular cases. We present in Part II the first classification of pencils of quadrics according to the real type of the intersection and we show how this classification can be used to efficiently determine the type of the real part of the intersection of two arbitrary quadrics. This classification is at the core of the design of our algorithms for computing near-optimal parameterizations of the real part of the intersection in all singular cases. We present these algorithms in Part III and give examples covering all the possible situations in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients.
Url:
Links toward previous steps (curation, corpus...)
- to stream Hal, to step Corpus: 003508
- to stream Hal, to step Curation: 003508
- to stream Hal, to step Checkpoint: 004902
Links to Exploration step
Hal:inria-00071229Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Near-Optimal Parameterization of the Intersection of Quadrics: I. The Generic Algorithm</title>
<author><name sortKey="Dupont, Laurent" sort="Dupont, Laurent" uniqKey="Dupont L" first="Laurent" last="Dupont">Laurent Dupont</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
<author><name sortKey="Lazard, Daniel" sort="Lazard, Daniel" uniqKey="Lazard D" first="Daniel" last="Lazard">Daniel Lazard</name>
<affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-233" status="VALID"><idno type="RNSR">199712651U</idno>
<orgName>Laboratoire d'Informatique de Paris 6</orgName>
<orgName type="acronym">LIP6</orgName>
<desc><address><addrLine>4 Place JUSSIEU 75252 PARIS CEDEX 05</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.lip6.fr/</ref>
</desc>
<listRelation><relation active="#struct-93591" type="direct"></relation>
<relation name="UMR7606" active="#struct-441569" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-93591" type="direct"><org type="institution" xml:id="struct-93591" status="VALID"><orgName>Université Pierre et Marie Curie - Paris 6</orgName>
<orgName type="acronym">UPMC</orgName>
<desc><address><addrLine>4 place Jussieu - 75005 Paris</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.upmc.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7606" active="#struct-441569" type="direct"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
</affiliation>
</author>
<author><name sortKey="Lazard, Sylvain" sort="Lazard, Sylvain" uniqKey="Lazard S" first="Sylvain" last="Lazard">Sylvain Lazard</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
<author><name sortKey="Petitjean, Sylvain" sort="Petitjean, Sylvain" uniqKey="Petitjean S" first="Sylvain" last="Petitjean">Sylvain Petitjean</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">HAL</idno>
<idno type="RBID">Hal:inria-00071229</idno>
<idno type="halId">inria-00071229</idno>
<idno type="halUri">https://hal.inria.fr/inria-00071229</idno>
<idno type="url">https://hal.inria.fr/inria-00071229</idno>
<date when="2005">2005</date>
<idno type="wicri:Area/Hal/Corpus">003508</idno>
<idno type="wicri:Area/Hal/Curation">003508</idno>
<idno type="wicri:Area/Hal/Checkpoint">004902</idno>
<idno type="wicri:explorRef" wicri:stream="Hal" wicri:step="Checkpoint">004902</idno>
<idno type="wicri:Area/Main/Merge">006813</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">Near-Optimal Parameterization of the Intersection of Quadrics: I. The Generic Algorithm</title>
<author><name sortKey="Dupont, Laurent" sort="Dupont, Laurent" uniqKey="Dupont L" first="Laurent" last="Dupont">Laurent Dupont</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
<author><name sortKey="Lazard, Daniel" sort="Lazard, Daniel" uniqKey="Lazard D" first="Daniel" last="Lazard">Daniel Lazard</name>
<affiliation wicri:level="1"><hal:affiliation type="laboratory" xml:id="struct-233" status="VALID"><idno type="RNSR">199712651U</idno>
<orgName>Laboratoire d'Informatique de Paris 6</orgName>
<orgName type="acronym">LIP6</orgName>
<desc><address><addrLine>4 Place JUSSIEU 75252 PARIS CEDEX 05</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.lip6.fr/</ref>
</desc>
<listRelation><relation active="#struct-93591" type="direct"></relation>
<relation name="UMR7606" active="#struct-441569" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-93591" type="direct"><org type="institution" xml:id="struct-93591" status="VALID"><orgName>Université Pierre et Marie Curie - Paris 6</orgName>
<orgName type="acronym">UPMC</orgName>
<desc><address><addrLine>4 place Jussieu - 75005 Paris</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.upmc.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7606" active="#struct-441569" type="direct"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
</affiliation>
</author>
<author><name sortKey="Lazard, Sylvain" sort="Lazard, Sylvain" uniqKey="Lazard S" first="Sylvain" last="Lazard">Sylvain Lazard</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
<author><name sortKey="Petitjean, Sylvain" sort="Petitjean, Sylvain" uniqKey="Petitjean S" first="Sylvain" last="Petitjean">Sylvain Petitjean</name>
<affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2573" status="OLD"><idno type="RNSR">200518305E</idno>
<orgName>Effective Geometric Algorithms for Surfaces and Visibility</orgName>
<orgName type="acronym">VEGAS</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/vegas</ref>
</desc>
<listRelation><relation active="#struct-160" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-300291" type="indirect"></relation>
<relation active="#struct-300292" type="indirect"></relation>
<relation active="#struct-300293" type="indirect"></relation>
<relation active="#struct-2496" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation name="UMR7503" active="#struct-441569" type="direct"></relation>
<relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-300291" type="direct"></relation>
<relation active="#struct-300292" type="direct"></relation>
<relation active="#struct-300293" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle name="UMR7503" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="ISNI">0000000122597504</idno>
<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName>
<orgName type="acronym">UHP</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName>
<date type="end">2011-12-31</date>
<desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName>
<orgName type="acronym">INPL</orgName>
<date type="end">2011-12-31</date>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
<tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
<country>France</country>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Université Nancy 2</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
<placeName><settlement type="city">Nancy</settlement>
<region type="region" nuts="2">Grand Est</region>
<region type="old region" nuts="2">Lorraine (région)</region>
</placeName>
<orgName type="university">Institut national polytechnique de Lorraine</orgName>
<orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName>
</affiliation>
</author>
</analytic>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>CURVE PARAMETERIZATION</term>
<term>INTERSECTION OF SURFACES</term>
<term>PENCILS OF QUADRICS</term>
<term>QUADRICS</term>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">We present the first efficient algorithm for computing an exact parametric representation of the intersection of two quadrics in three-dimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection are rational functions whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, or two conics). Furthermore, the parameterization is near-optimal in the sense that the number of square roots appearing in the coefficients of these functions is minimal except in a small number of cases where there may be an extra square root. In addition, the algorithm is practical: a complete, robust and efficient C++ implementation is described in Part IV [12] of this paper. In Part I, we present an algorithm for computing a parameterization of the intersection of two arbitrary quadrics which we prove to be near-optimal in the generic, smooth quartic, case. Parts II and III [4, 5] treat the singular cases. We present in Part II the first classification of pencils of quadrics according to the real type of the intersection and we show how this classification can be used to efficiently determine the type of the real part of the intersection of two arbitrary quadrics. This classification is at the core of the design of our algorithms for computing near-optimal parameterizations of the real part of the intersection in all singular cases. We present these algorithms in Part III and give examples covering all the possible situations in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients.</div>
</front>
</TEI>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Main/Merge
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 006813 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Merge/biblio.hfd -nk 006813 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Lorraine |area= InforLorV4 |flux= Main |étape= Merge |type= RBID |clé= Hal:inria-00071229 |texte= Near-Optimal Parameterization of the Intersection of Quadrics: I. The Generic Algorithm }}
This area was generated with Dilib version V0.6.33. |