Apollonius Problem and Caustics of an Ellipsoid

Yıl 2024, Cilt: 17 Sayı: 2, 402 - 420, 27.10.2024

Yagub Aliyev

https://doi.org/10.36890/iejg.1368456

Öz

In the paper we discuss Apollonius Problem on the number of normals of an ellipse passing through a given point. It is known that the number is dependent on the position of the given point with respect to a certain astroida. The intersection points of the astroida and the ellipse are used to study the case when the given point is on the ellipse. The problem is then generalized for 3 dimensional space, namely for Ellipsoids. The number of concurrent normals in this case is known to be dependent on the position of the given point with respect to caustics of the ellipsoid. If the given point is on the ellipsoid then the number of normals is dependent on position of the point with respect to the intersections of the ellipsoid with its caustics. The main motivation of this paper is to classify all possible cases of these intersections.

Anahtar Kelimeler

Ellipsoid, Apollonius, centro-surface, caustics, astroida, ellipse

Kaynakça

• [1] Abbena, E., Salamon, S., Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. 3rd Edition. Chapman and Hall/CRC, New York (2017). https://doi.org/10.1201/9781315276038
• [2] Aliyev, Y.: Cayley’s Centro-Surface: Old and New Attempts to Draw this Elusive Surface and Some New Ideas Around It, In: Contributed Talks, Maple in Education, Maple Conference, Nov 02-04 (2022). https://www.maplesoft.com/mapleconference/2022/ full-program.aspx and https://youtu.be/x-lhM7oja_Q
• [3] Apollonius of Perga: Conics Books V to VII, The Arabic Translation of the Lost Greek Original in the Version of the Banu Musa, Gerald J. Toomer (ed.), Springer New York, NY (1990). https://doi.org/10.1007/978-1-4613-8985-9
• [4] Arnol’d, V. I.: Astroidal geometry of hypocycloids and the Hessian topology of hyperbolic polynomials. Russian Math. Surveys 56(6), 1019–1083 (2001). https://doi.org/10.1070/RM2001v056n06ABEH000452
• [5] Arnold, V. I.: Astroidal geometry and topology (in Russian), Summer school "Contemporary Mathematics", July 15-27, (2001). Lecture notes: https://www.mccme.ru/dubna/2001/material/arnbook.pdf Video recordings: 6 videos https://youtube.com/ playlist?list=PLdjLKCP6MkB0HAsjuDQcJvUbz9IhsXQuG
• [6] Arnold, V. I.: Catastrophe Theory. Springer (1992). https://doi.org/10.1007/978-3-642-58124-3
• [7] Afrajmovich, V. S., Il’yashenko, Yu. S., Shil’nikov, L. P., Arnold, V. I.: Dynamical Systems V. Bifurcation Theory and Catastrophe Theory. Encyclopaedia of Mathematical Sciences. Springer (1994). https://doi.org/10.1007/978-3-642-57884-7
• [8] Banchoff, T.: Differential geometry and computer graphics. Perspectives in Mathematics. Anniversary of Oberwolfach. Birkhauser Verlag. Basel, 43-60 (1984).
• [9] Banchoff, T. F.: Beyond the Third Dimension, Geometry, Computer Graphics, and Higher Dimensions, Scientific American Library, W H Freeman & Co (1990). Online version: https://www.math.brown.edu/tbanchof/Beyond3D.new/chapter7/s7_8.html
• [10] Bektas, S.: Geodesy II Applications on the surface of an ellipsoid (Turkish). Atlas Akademi. Konya (2021). https://www.atlasakademiyayin.com/Stok/StokDetay/26071
• [11] Berger, M.: Geometrie. Cedic/ Fernand Nathan. Paris (1978).
• [12] Berger, M.: Geometry Revealed A Jacob’s Ladder to Modern Higher Geometry. Springer New York. NY (2010). https://doi.org/10. 1007/978-3-540-70997-8
• [13] Berger, M., Pansu, P., Berry, J., Saint-Raymond, X: Problems in Geometry. Springer. New York (1984). https://doi.org/10.1007/ 978-1-4757-1836-2
• [14] Brill, L.: Central Surface of a Paraboloid. Geometric Model. No. 149. Ser. 1, No. 2a, The National Museum of American History (1892). https://americanhistory.si.edu/collections/search/object/nmah_693993
• [15] Caspari, F.: Die Krümmungsmittelpunktsfläche des elliptischen Paraboloids. Dissert. Reimer. Berlin (1875). http://resolver.sub. uni-goettingen.de/purl?PPN310966825
• [16] Cayley, A.: Note sur les normales d’une conique. Journal für die reine und angewandte Mathematik 56, 182-185 (1859). http://resolver. sub.uni-goettingen.de/purl?PPN243919689_0056
• [17] Cayley, A.: On the centro-surface of an ellipsoid. Transactions of the Cambridge Philosophical Society 12(1), 319-365 (1873). Also included in The collected mathematical papers of Arthur Cayley, Vol. VIII, Cambridge University Press, Cambridge, 316-365 (1895). http://name. umdl.umich.edu/ABS3153.0008.001
• [18] Cherrie, M. A.: The conjugate locus in convex manifolds. Doctoral Thesis. School of Mathematics & Physics. University of Portsmouth (2022). https://researchportal.port.ac.uk/en/studentTheses/the-conjugate-locus-in-convex-manifolds
• [19] Getty Museum Collection. Centro-Surface. Ellipsoid. about 1860. https://www.getty.edu/art/collection/object/107CZM
• [20] Clebsch, A.; Ueber das Problem der Normalen bei Curven und Oberflächen der zweiten Ordnung. Journal für die reine und angewandte Mathematik 62, 64-109 (1863). http://eudml.org/doc/147884
• [21] Desboves, A.: Théorèmes et problèmes sur les normal. Exercices pour les classes de mathématiques spéciales. Mallet-Bachelier. Paris (1861). https://books.google.az/books?id=BVwGZKiMM7wC&pg=PP1#v=onepage&q&f=false
• [22] Domokos, G., Lángi, Z., Szabó, T.: A topological classification of convex bodies, Geom. Dedicata 182, 95–116 (2016). https://doi.org/10. 1007/s10711-015-0130-4
• [23] Drach, K. D., Komlev, V.: Space evolute of an elliptic paraboloid and a one-sheeted hyperboloid of M. Schilling catalogue. Category: Space caustics of quadrics. Geometric Models Collection of V.N. Karazin Kharkiv National University. http://touch-geometry.karazin. ua/list/category-space-caustics-of-quadrics
• [24] Dyck W. (ed.), Katalog mathematischer und mathematisch-physikalischer modelle, apparate und instrumente. Unter mitwirkung zahlreicher fachgenossen. Deutsche Mathematiker-Vereinigung. C. Wolf & Sohn. München (1892). https://archive.org/details/ katalogmathemat00goog/page/281/mode/2up
• [25] Dyck, W.: Die Centralfläche des einschaligen Hyperboloids. Abhandlungen und Erläuterungen zu den mathematischen Modellen der Serien I-XII des Modellverlags, unter Leitung von L. Brill, 13-18, Darmstadt (1877-1885). https://opendigi.ub.uni-tuebingen.de/ opendigi/BRILL#tab=struct&p=21
• [26] Eisenhart, L. P.: An introduction to differential geometry: with use of the tensor calculus. Princeton University Press. Princeton (1947). https://archive.org/details/introductiontodi0000eise/mode/2up
• [27] European Space Agency, Seeing quintuple. Press release in Science & Exploration. Space Science. Hubble Space Telescope. Aug 13, (2021). https://www.esa.int/ESA_Multimedia/Images/2021/08/Seeing_quintuple Also posted as Hubble Sees Cosmic Quintuple. Solar System and Beyond. NASA Hubble Mission Team. Goddard Space Flight Center. Aug 27, (2021). https://www.nasa.gov/ image-feature/goddard/2021/hubble-sees-cosmic-quintuple
• [28] Ferguson, C. C.: Intersections of ellipsoids and planes of arbitrary orientation and position. Mathematical Geology 11, 329-336 (1979). https: //doi.org/10.1007/BF01034997
• [29] Fischer G. (ed.): Mathematical Models. From the Collections of Universities and Museums. Photograph Volume and Commentary. Springer Spektrum (2017). https://doi.org/10.1007/978-3-658-18865-8
• [30] Forsyth, A. R.: Lectures on the differential geometry of curves and surfaces. Cambridge University Press (1912). https://archive. org/details/cu31924060289141/page/112/mode/2up
• [31] Gendzwill, D. J., Stauffer, M.R.: Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections. Mathematical Geology 13, 135–152 (1981). https://doi.org/10.1007/BF01031390
• [32] Geiser, C. F.: La normali dell’essoide, Annali Di Matematica Pura Ed Applicata 1, 317-328 (1867). https://ia800708.us.archive. org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1007%252Fbf02360180. zip&file=10.1007%252Fbf02419181.pdf https://doi.org/10.1007/BF02419181
• [33] Göttinger Sammlung mathematischer Modelle und Instrumente. Curvature centre point surface of the hyperboloid of one sheet, the triaxial ellipsoid and the elliptic paraboloid. Krümmungsmittelpunktsfläche Modellen: 238, 239, 242 und 345, Gypsum; Georg-August Universität Göttingen. https://sammlungen.uni-goettingen.de/objekt/record_DE-MUS-069123_238/1/-/ https://sammlungen.uni-goettingen.de/objekt/record_DE-MUS-069123_242/1/-/ https://sammlungen.uni-goettingen.de/objekt/record_DE-MUS-069123_239/1/-/ https://sammlungen.uni-goettingen.de/objekt/record_DE-MUS-069123_345/1/-/
• [34] Hamflett,W.G.2014.Joachimsthal’s Theorem. The Mathematical Gazette 329 (299), 86- 87 (1948). https://doi.org/10.2307/3610710
• [35] Hartmann, F., Jantzen, R.: Apollonius’s Ellipse and Evolute Revisited. Convergence. August (2010). https://www.maa.org/book/ export/html/116798
• [36] Itoh, Ji., Kiyohara, K.: The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds. Arnold Math J. 7, 31–90 (2021). https://doi.org/10.1007/s40598-020-00153-9
• [37] Joachimsthal, F.: Über die Normalen der Ellipse und des Ellipsoids. Journal für die reine und angewandte Mathematik 26, 172–178 (1843). http://resolver.sub.uni-goettingen.de/purl?GDZPPN002143445
• [38] Joachimsthal, F.: Ueber die Anzahl reeller Normalen, welche von einem Punkte an ein Ellipsoid gezogen werden können. Journal für die reine und angewandte Mathematik 59, 111-124 (1861). https://doi.org/10.1515/crll.1861.59.111 https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0059/PPN243919689_0059.pdf
• [39] Joets, A., Ribotta, R.: Caustique de la surface ellipsoïdale à trois dimensions. Experimental Mathematics 8(1), 49 55 (1999). https://projecteuclid.org/journals/experimental-mathematics/volume-8/issue-1/ Caustique-de-la-surface-ellipso%C3%AFdale-%C3%A0-trois-dimensions/em/1047477111.full
• [40] Junker, H.: Anschauungsmodelle in der mathematischen Forschung deutscher Gelehrter 1860–1877. Dissert. Martin-Luther Universität Halle-Wittenberg (2023). https://opendata.uni-halle.de/bitstream/1981185920/110975/1/Dissertation_ MLU_2023_JunkerHannes.pdf
• [41] Kagan, V. F.: Foundations of the Theory of Surfaces (in Russian). Part 1. OGIZ. Moscow (1947).
• [42] Kummer, E.: Über ein Modell der Krümmungsmittelpunktsfläche des dreiaxigen Ellipsoids, Sitzung der physikalisch-mathematischen Klasse vom 30. Juni 1862. Monatsberichten der königlichen preussischen Akademie der Wissenschaften 426-428 (1862). https://www. biodiversitylibrary.org/page/36507229#page/464/mode/1up
• [43] Lando, S. K.: Something about Caustics (in Russian), Summer School "Contemporary Mathematics". July 18–29. Dubna (2015). http: //www.mathnet.ru/present12099 https://www.mccme.ru/dubna/2015/courses/lando.html
• [44] Legendre, A. M.: Traité des fonctions elliptiques et des intégrales eulériennes: avec des tables pour en faciliter le calcul numérique. Vol. 1. Huzard-Courcier. Paris (1825). https://books.google.az/books?id=IC_vAAAAMAAJ&pg=PA350&source=gbs_toc_r&cad= 3#v=onepage&q&f=false
• [45] Milne, J. J.: The Conics of Apollonius. The Mathematical Gazette 1(6), 49-55 (1895). https://doi.org/10.2307/3604717
• [46] Monge, G.: Application de l’Analyse à la Géométrie (5e édition). Paris (1850). https://gallica.bnf.fr/ark:/12148/ bpt6k96431405/f344.item
• [47] Müller, R.: Ueber eine gewisse Gleichung 2n-ten Grades deren Specialfälle n = 2 und n = 3 beim Normalenproblem der Ellipse und des Ellipsoides auftreten. Dissert. G. Schade. Berlin (1884). https://edoc.hu-berlin.de/bitstream/handle/18452/786/27001. pdf?sequence=1
• [48] Nádeník, Z., Beˇ cváˇr, J.: Moji uˇ citelé geometrie (In Czech). Matfyzpress. Praha (2011). http://dml.cz/dmlcz/402172
• [49] Niemtschik, R.: Einfaches Verfahren, Normalen zu Flächen zweiter Ordnung durch ausserhalb liegende Punkte zu ziehen. Sitzungsberichte der math.-naturwiss. Classe der kaiserlichen Akademie der Wissenschaften Wien 58 II. Abtheilung. 831–836 (1868). https://books. google.az/books?id=s3M5AAAAcAAJ&pg=PP1#v=onepage&q&f=false
• [50] Nikolsky, S. M.: A Course of Mathematical Analysis. Vol. 1. Mir Publishers. Moscow (1977). https://archive.org/details/ nikolsky-a-course-of-mathematical-analysis-vol-1-mir/page/203/mode/2up
• [51] O’Neill, B.: Elementary Differential Geometry. Second Edition. Academic Press 2006. https://doi.org/10.1016/ C2009-0-05241-6
• [52] Salmon, G.: Ontheequation of the surface of centres of an ellipsoid. The Quarterly Journal of Pure And Applied Mathematics 2, 217–222 (1858). https://babel.hathitrust.org/cgi/pt?id=uc1.$b417524&view=1up&seq=241
• [53] Schmidt, T., Treu T. et al., STRIDES: automated uniform models for 30 quadruply imaged quasars. Monthly Notices of the Royal Astronomical Society 518(1), 1260–1300 (2023). https://doi.org/10.1093/mnras/stac2235
• [54] Schröder, H.: Die Zentraflächen der Paraboloide und Mittelpunktsflächen zweiten Grades. Dissert. H. John. Halle (1913). http: //resolver.sub.uni-goettingen.de/purl?PPN316295612
• [55] Tikhomirov, V. M.: Stories about maxima and minima. Mathematical world series 1. AMS (1990). https://bookstore.ams.org/ view?ProductCode=MAWRLD/1
• [56] Trocado, A., Gonzalez-Vega, L., Dos Santos, J.M.: Intersecting Two Quadrics with GeoGebra. In: ´ Ciri´ c, M., Droste, M., Pin, JÉ. (eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer Science (LNCS), vol 11545. Springer, Cham, 2019, 237-248. https: //doi.org/10.1007/978-3-030-21363-3_20
• [57] Tsar’kov, I. G.: Mach Disks and Caustic Reflections, Caustics, Application to Astrophysics. International Online Conference "Mathematical Physics, Dynamical Systems and Infinite-Dimensional Analysis"-MPDSIDA. Book of Abstracts. 201-202 (2021). https://www. mathnet.ru/supplement/conf/1918/Abstracts-MPDSIDA2021.pdf
• [58] Tsarkov, I. G.: Mach discs and caustic reflections, caustics, application in astrophysics. Proceedings of international scientific conference dedicated to 70th birthday of academician NAS of Tajikistan Shabozov M.Sh. Modern problems of mathematical analysis and theory of functions. OOO ER-GRAF. Dushanbe. 138-144 (2022).
• [59] Van Der Waerden, B. L.: Science Awakening I, Springer Netherlands (1975). https://link.springer.com/book/10.1007/ 978-94-009-1379-0
• [60] Weatherburn, C. E.: Differential Geometry Of Three Dimensions. Vol.1. Cambridge University Press (1955). https://archive.org/ details/differentialgeom003681mbp/page/n175/mode/2up
• [61] Yang, W.-Ch. Lo, M.-L: General Inverses in 2-D, 3-D, applications inspired by Technology, The Electronic Journal of Mathematics and Technology 2(3), 243-260 (2008). https://php.radford.edu/~ejmt/
• [62] Yoshizawa, S., Belyaev, A., Yokota, H., Seidel, H.-P.: Fast and Faithful Geometric Algorithm for Detecting Crest Lines on Meshes. 15th Pacific Conference on Computer Graphics and Applications (PG’07). Maui. HI. USA. 231-237 (2007). https://doi.org/10.1109/PG.2007. 24
• [63] Zel’dovich, Ya. B., Mamaev, A.V., Shandarin, S.F.: Laboratory observation of caustics, optical simulation of the motion of particles, and cosmology, Phys. Usp. 26(1), 77-83 (1983). https://doi.org/10.1070/PU1983v026n01ABEH004307
• [64] Zel’dovich, Ya. B., Myskis, A. D.: Elements of Mathematical Physics. A medium of non-interacting particles (in Russian). Fizmatlit. Moscow (2008).