[Cyclides de Darboux translationnelles et Grandes cyclides de Darboux]
Une surface qui est la somme ponctuelle de cercles dans l’espace euclidien est soit coplanaire, soit ne contient pas plus de 2 cercles passant par un point général. Une surface qui est le produit ponctuel de cercles dans les quaternions unitaires contient soit 2, 3, 4, ou 5 cercles passant par un point général. Une surface dans une sphère unitaire de n’importe quelle dimension qui contient 2 grands cercles passant par un point général contient soit 4, 5, 6, ou une infinité de cercles passant par un point général. Ce sont quelques corollaires de notre classification des cyclides de translation et des cyclides de Darboux. Nous utilisons la combinatoire associée à l’ensemble des courbes de faible degré sur de telles surfaces modulo l’équivalence numérique.
A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2, 3, 4, or 5 circles through a general point. A surface in a unit-sphere of any dimension that contains 2 great circles through a general point contains either 4, 5, 6, or infinitely many circles through a general point. These are some corollaries from our classification of translational and great Darboux cyclides. We use the combinatorics associated to the set of low degree curves on such surfaces modulo numerical equivalence.
Révisé le :
Accepté le :
Publié le :
Keywords: real surfaces, pencils of circles, singular locus, Darboux cyclides, Clifford torus, Möbius geometry, elliptic geometry, hyperbolic geometry, Euclidean geometry, Euclidean translations, Clifford translations, unit quaternions, weak del Pezzo surfaces, divisor classes, Néron–Severi lattice
Mot clés : surfaces réelles, faisceaux de cercles, lieu singulier, cyclides de Darboux, tore de Clifford, géométrie de Möbius, géométrie elliptique, géométrie hyperbolique, géométrie euclidienne, translations euclidiennes, translations de Clifford, quaternions unitaires, surfaces de del Pezzo faibles, classes de diviseurs, réseau de Néron–Severi
Niels Lubbes 1
CC-BY 4.0
@article{CRMATH_2024__362_G4_413_0,
author = {Niels Lubbes},
title = {Translational and great {Darboux} cyclides},
journal = {Comptes Rendus. Math\'ematique},
pages = {413--448},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.603},
language = {en},
}
Niels Lubbes. Translational and great Darboux cyclides. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 413-448. doi : 10.5802/crmath.603. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.603/
[1] On the parameterization of rational ringed surfaces and rational canal surfaces, Math. Comput. Sci., Volume 8 (2014) no. 2, pp. 299-319 | DOI | MR
[2] Circles on surfaces in the Euclidean -space, Geometry and differential geometry (Proc. Conf., Univ. Haifa, 1979) (Lecture Notes in Mathematics), Volume 792, Springer, 1980, pp. 213-221 | DOI | MR | Zbl
[3] A Treatise on the Circle and Sphere, Oxford University Press, 1916, 603 pages
[4] Non-Euclidean geometry, Spectrum, MAA, 1998, xviii+336 pages | DOI
[5] Sur le contact des coniques et des surfaces, C. R. Acad. Sci. Paris (1880) no. 91, pp. 969-971
[6] Classical algebraic geometry: A modern view, Cambridge University Press, 2012, xii+639 pages | DOI
[7] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | DOI
[8] Surfaces with orthogonal families of circles, Proc. Am. Math. Soc., Volume 123 (1995) no. 3, pp. 865-872 | DOI | MR
[9] Nicht-Euklidische Geometrie, II, Vorlesung, Göttingen, 1893
[10] Lectures on resolution of singularities, 166, Princeton University Press, 2007, vi+208 pages
[11] Über die Flächen vierten Grades, auf welchen Schaaren von Kegelschnitten liegen, J. Reine Angew. Math., Volume 64 (1863) no. 11, pp. 66-76 | Zbl
[12] Rational Bezier formulas with quaternion and Clifford algebra weights, SAGA - Advances in ShApes, Geometry, and Algebra, Geometry and Computing, Volume 10 (2014), pp. 147-166 | DOI | Zbl
[13] Design of a Variable-Mobility Linkage Using the Bohemian Dome, J. Mech. Des., Volume 141 (2019) no. 9, 092303, 12 pages
[14] Kinematic generation of Darboux cyclides, Comput.-Aided Geom. Des., Volume 64 (2018), pp. 11-14 | DOI | MR | Zbl
[15] Orbital. Sage library for constructing and visualizing curves on surfaces, 2017 (https://github.com/niels-lubbes/orbital)
[16] Surfaces that are covered by two pencils of circles, Math. Z., Volume 299 (2021) no. 3-4, pp. 1445-1472 | DOI | MR
[17] Cyclides, 2023 (https://github.com/niels-lubbes/cyclides)
[18] Surfaces containing two isotropic circles through each point, Comput.-Aided Geom. Des., Volume 90 (2021), 102035, 15 pages | MR
[19] Architectural Geometry, Bentley Institute Press, 2007, 724 pages
[20] Generalized Dupin Cyclides with Rational Lines of Curvature, Curves and surfaces (Lecture Notes in Computer Science), Volume 6920, Springer, 2012, pp. 543-552 | DOI | MR | Zbl
[21] Darboux cyclides and webs from circles, Comput.-Aided Geom. Des., Volume 29 (2012) no. 1, pp. 77-97 | DOI | MR | Zbl
[22] The multiple conical surfaces, Beitr. Algebra Geom., Volume 42 (2001) no. 1, pp. 71-87 | MR
[23] Topics in Galois theory, Jones and Bartlett Publishers, 1992, xvi+117 pages
[24] Real algebraic surfaces, Lecture Notes in Mathematics, 1392, Springer, 1989, x+215 pages | DOI | MR
[25] Surfaces containing two circles through each point, Math. Ann., Volume 373 (2018) no. 3–4, pp. 1299-1327 | MR
[26] Rational motions with generic trajectories of low degree, Comput.-Aided Geom. Des., Volume 76 (2020), 101793, 10 pages | MR
[27] Cyclides, Hokkaido Math. J., Volume 29 (2000) no. 1, pp. 119-148 | MR
[28] A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil, Linear Algebra Appl., Volume 14 (1976), pp. 189-209 | DOI | MR | Zbl
[29] Théorème sur le tore, Nouv. Ann. Math., Volume 7 (1848), pp. 345-347 (https://eudml.org/doc/95880)
[30] Enumerating the morphologies of non-degenerate Darboux cyclides, Comput.-Aided Geom. Des., Volume 75 (2019), 101776, 15 pages | MR | Zbl
[31] Sage Mathematics Software (2012) (https://www.sagemath.org)
Cité par Sources :
Commentaires - Politique