, ensemble des résultats de complexité algébriqué enoncés dans les sections précédentes Quand le résultat théorique n'est pas optimal, nous indiquons la nature du test qu'il faudrait effectuer pour vérifier, en pratique, si le résultat obtenu est optimal ou s'il contient une racine carrée non nécessaire. Notons qu'hormis le cas o` u la caractéristique de Segre est, et dans ce cas nous obtenons toujours le résultat optimal)
, il existe des faisceaux ne contenant pas de quadriquessingulì eres De plus, pour les quadriques de rang maximal, la situation se complique nécessairement en augmentant la dimension de l'espace puisque dans P 3 (R), le choix des inerties possibles se limitè a, alors que dans les cas général, le nombre de cas possibles augmente. Nous devons donc nous poser deux questions, pour chaque espace P n (R), existe-t-il des quadriques de rang n + 1, pp.2-4
, il existe (u, v) et (s, t) dans P 1 (R) tels que
, Posons enfin v = W , et nous avons bien trouvé, 1) deux points de P 1 (R) tels que (X, Y, Z, W ) = (ut, vs, us, vt) et XY = ZW . Supposons maintenant que X = 0. L'´ equation XY = ZW implique que Z = 0 ou W = 0. Supposons que Z = 0 (le cas W = 0 est identique par la symétrie de l'expression) Nous avons alors (ut, vs, us, vt) = (0, Y, 0, W ). v = 0 car sinon (ut, vs, us, vt) seraitégaìseraitégaì a (0, 0, 0, 0) ? P 3 (R) Nous pouvons donc poser v = 1, Du coup, issue.0
, 3D modeling, animation, post-production
, Quadratic Forms and Their Classification by Means of Invariant Factors, Cambridge Tracts in Mathematics and Mathematical Physics, 1906.
, Exact collision detection of two moving ellipsoids under rational motions, Proc. of IEEE Conference on Robotics and Automation, 2003.
, Galois theory, Abstract Algebra, pp.471-570, 1998.
, IntroductionàIntroductionà la géométrie, 2002.
, ¨ Uber das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen, Comment. Math. Helv, vol.9, pp.188-192, 1936.
, Automatic parsing of degenerate quadric-surface intersections, ACM Transactions on Graphics, vol.8, issue.3, pp.174-203, 1989.
DOI : 10.1145/77055.77058
, Computing a 3-dimensional cell in an arrangement of quadrics : exactly and actually, Proc. of ACM Symposium on Computational Geometry, pp.264-273, 2001.
, The convex hull of ellipsoids, Proceedings of the seventeenth annual symposium on Computational geometry , SCG '01, pp.321-322, 2001.
DOI : 10.1145/378583.378717
, Combining algebraic rigor with geometric robustness for the detection and calculation of conic sections in the intersection of two natural quadric surfaces, Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications , SMA '91, pp.221-231, 1991.
DOI : 10.1145/112515.112545
, The intersection of two ruled surfaces, Computer-Aided Design, vol.31, issue.1, pp.33-50, 1999.
DOI : 10.1016/S0010-4485(98)00078-5
, Constructing convex hulls of quadratic surface patches
, Geometric and Solid Modeling : An Introduction, 1989.
, PRECISE : Efficient Multiprecision Evaluation of Algebraic Roots and Predicates for Reliable Geometric Computation, ACM Symposium on computational Geometry, 2001.
, Efficient and accurate b-rep generation of low degree sculptured solids using exact arithmetic, Symposium on Solid Modeling and Applications, pp.42-55, 1997.
, A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces, Communications of the ACM, vol.19, issue.10, pp.555-563, 1976.
, Mathematical models for determining the intersections of quadric surfaces, Computer Graphics and Image Processing, vol.11, issue.1, pp.73-87, 1979.
, Quadril : A computer language for the description of quadric-surface bodies, Proceedings of the 7th annual conference on Computer graphics and interactive techniques, pp.86-92, 1980.
, Intersecting quadrics, Proceedings of the twentieth annual symposium on Computational geometry , SCG '04, pp.419-428, 2004.
DOI : 10.1145/997817.997880
URL : https://hal.archives-ouvertes.fr/inria-00104003
, Geometric algorithms for detecting and calculating all conic sections in the intersection of any two natural quadric surfaces, Graphical Models and Image Processing, vol.57, issue.1, pp.55-66, 1995.
, Geometric approaches to nonplanar quadric surface intersection curves, ACM Transactions on Graphics, vol.6, issue.4, pp.274-307, 1987.
, Geometry of Projective Algebraic Curves, 1984.
, Solid Modeling: A Historical Summary and Contemporary Assessment, IEEE Computer Graphics and Applications, vol.2, issue.2, pp.9-24, 1982.
DOI : 10.1109/MCG.1982.1674149
, Efficient isolation of polynomial's real roots, Journal of Computational and Applied Mathematics, vol.162, issue.1, pp.33-50, 2004.
DOI : 10.1016/j.cam.2003.08.015
, Algebraic methods for intersections of quadric surfaces in GMSOLID, Computer Vision, Graphics, and Image Processing, vol.22, pp.222-238, 1983.
, Studio sulle quadriche in uno spazio lineare ad un numero qualunque di dimensioni, 1883.
, On the planar intersection of natural quadrics, Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications , SMA '91, pp.233-242, 1991.
DOI : 10.1145/112515.112546
, Computing the intersection of a plane and a natural quadric, Computer & Graphics, vol.12, issue.2, pp.179-186, 1992.
, Computing the intersection of a plane and a revolute quadric, Computer & Graphics, vol.18, issue.1, pp.47-59, 1994.
, On the lower degree intersections of two natural quadrics, ACM Transactions on Graphics, vol.13, issue.4, pp.400-424, 1994.
, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, Philosophical Magazine, vol.4, pp.138-142
, Classifying the Nonsingular Intersection Curve of Two Quadric Surfaces, Proceedings of the Geometric Modeling and Processing and Theory and Applications, p.23, 2002.
, Definite and semidefinite matrices in a real symmetric matrix pencil, Pacific Journal of Mathematics, vol.49, pp.561-568, 1973.
, A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil, Linear Algebra and Its Applications, vol.14, pp.189-209, 1976.
, A recurring theorem about pairs of quadratic forms and extension : a survey, Linear Algebra and Its Applications, vol.25, pp.219-237, 1979.
, Efficient collision detection for moving ellipsoids based on simple algebraic test and separating planes, 2003.
, Zur Theorie der bilinearen und quadratischen Formen, Monatshefte Akademie der Wissenschaften, pp.310-338, 1868.
, Enhancing Levin's method for computing quadric-surface intersections, Computer Aided Geometric Design, vol.20, issue.7, pp.401-422, 2003.
DOI : 10.1016/S0167-8396(03)00081-5
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.600
, Rational Quadratic Parameterizations of Quadrics, International Journal of Computational Geometry & Applications, vol.07, issue.06, pp.599-619, 1997.
DOI : 10.1142/S0218195997000375
, Computing quadric surface intersections based on an analysis of plane cubic curves, Graphical Models, vol.64, issue.6, pp.335-367, 2002.
DOI : 10.1016/S1077-3169(02)00018-7
, Quadric-surface intersection curves: shape and structure, Computer-Aided Design, vol.25, issue.10, pp.633-643, 1993.
DOI : 10.1016/0010-4485(93)90018-J
, An algebraic condition for the separation of two ellipsoids, Computer Aided Geometric Design, vol.18, issue.6, pp.531-539, 2001.
DOI : 10.1016/S0167-8396(01)00049-8