ÇOK MERKEZLİ KAPALI BİR EĞRİ: CASSİNİ OVALİ, ÖZELLİKLERİ VE UYGULAMALARI

Yıl 2013, Cilt: 14 Sayı: 2, 231 - 248, 01.07.2013

Mümtaz Karataş

Öz

Cassini ovali, düzlem üzerindeki sabit iki noktaya olan mesafelerinin çarpımı yine bir sabit olan noktalar kümesinin oluşturduğu kuadratik bir eğri olarak tanımlanabilir. Benzersiz özellikleri ve geometrik profili, bu ovalleri pek çok askeri ve ticari alanda kullanılmasına imkan tanıyan üstün bir araç haline getirmiş, ayrıca analitik geometriye ve genel elips konseptinin ötesindeki matematik teorisiyle ilişkili diğer konulara yeni bir boyut kazandırmıştır. Bu çalışma kapsamında, Cassini ovallarinin çeşitli özellikleriyle ilgili analitik ifadeler geliştirilmiş ve farklı alanlardan örnekler kullanılarak söz konusu ovallerin uygulama alanlarına ilişkin özet bilgi verilmiştir

Anahtar Kelimeler

Cassini Ovalleri, Çok Merkezli Eğriler

A MULTI FOCI CLOSED CURVE: CASSINI OVAL, ITS PROPERTIES AND APPLICATIONS

Öz

A Cassini oval is a quartic plane curve defined as the set or locus of points in the plane such that the product of the distances to two fixed points is constant. Its unique properties and miraculous geometrical profile make it a superior tool to utilize in diverse fields for military and commercial purposes and add new dimensions to analytical geometry and other subjects related to mathematics beyond the prevailing concept of ellipse. In this study we explore and derive analytical expressions for the properties of these curves and give a summary of its applications with distinct examples.

Anahtar Kelimeler

Cassini Ovals, Multi Foci Curves

Kaynakça

• ALLEN P.S., GROVE S., ZANCHE N. (2002). Nuclear magnetic resonance birdcage coil with Cassinian oval former. United States Patent, US 6,452,393 B1.
• ANGELOV B., MLADENOV I.M. (2000). On the geometry of red blood cell. International Conference on Geometry, Integrability and Quantization, Coral Press, Sofia, pp.27-46.
• BEISER A. (1997). Concept of modern physics. Tata McGraw Hill Publishing Company Ltd. N. Delhi. pp.131.
• CORALUPPI S., GRIMMETT D. (2003). Multistatic sonar tracking. Signal Processing, Sensor Fusion, and Target Recognition XII, I KADAR (ed.), Proceedings of SPIE Vol. 5096.
• COX, H. (1989). Fundamentals of bistatic active sonar. Underwater Acoustic Data Processing (Y. CHAN, ed.), Kluwer, pp.3-24.
• DAUKANTIENE V., PAPRECKIENE L., GUTAUSKAS M. (2003). Simulation and application of the behaviour of a textile fabric while pulling it through a round hole. Fibres & Textiles in Eastern Europe, Vol 11-2, pp.41.
• DI BIASIO A., CAMETTI C. (2005). Effect of the shape of human erythrocytes on the evaluation of the passive electrical properties of the cell membrane. Bioelectrochemistry. Vol 65-2, pp.163-169.
• GIBSON K. (2007). The ovals of Cassini. Lecture Notes.
• GILLESPIE C.C. (1971). Dictionary of scientific biography. New York: Charles Scribner’s Sons.
• GLENN J.A., LITTLER G. H. (1984). A dictionary of mathematics. London: Harper and Row.
• GUANGXI D., XIMING L. (1989). Approach to dynamics of fission described by Cassinian ovaloid. High Energy Physics and Nuclear Physics Journal, Vol 13-55, pp.8-455. Hash function. (2012). [Available at:] http://en.wikipedia.org/wiki/hash_function. [Accessed: 10.06.2012].
• HELLMERS J., EREMINA E., WRIEDT T. (2006). Simulation of light scattering by biconcave Cassini ovals using the nullfield method with discrete sources. Journal of Optics A: Pure and Applied Optics, Pure Appl. Opt. Vol 8. pp. 1-9. HIRST A. E., LLOYD E. K. (1997). Cassini, his ovals, and a space probe to Saturn. The Mathematical Gazette, Vol 81, No. 492, pp.409-421.
• JAMES G., JAMES R.C. (1949). Mathematics dictionary. D. Van Nostrand Co., Inc., New York. pp.39.
• JOHNSEN T., OLSEN K.E. (2006). Bi and multistatic radar. Advanced Radar Signal and Data Processing, Vol. 4-1, pp.4-34.
• KARATAŞ, M. (2012). Optimization of distributed underwater sensor networks with mixed integer non-linear programming. Marmara University Journal of Science, Vol. 24(3), pp.77-92.
• KHILJI M.J (2004). Multi foci closed curves. Journal of Theoretics, Vol 6-6.
• KOENDERINK J.J. (1990). Solid shape (Artificial intelligence). Massachusetts Institute of Technology, The MIT Press.
• MATZ F. (1985). The Rectification of the Cassinian Oval by Means of Elliptic Functions, Am.Math. Monthly, Vol 2, pp.221-357.
• MAZERON P., MULLER S. (1998). Dielectric or absorbing particles: EM surface fields and scattering. Journal of Optics, Vol 29, pp.68-77.
• PASHKEVICH V.V. (1971). On the asymmetric deformation of fissioning nuclei. Nuclear Physics Journal, Vol 169-2, pp.275–293.
• RAY K.S., RAY B.K. (2011). A method of deviation for drawing implicit curves. International Journal of Computer Graphics, Vol 2, No 2.
• RUSINOL M., LLADOS J. (2008). A region-based hashing approach for symbol spotting in technical documents, graphics recognition. Recent Advances and New Opportunities, pp.104-113.
• RYAN R., VERDERAIME V. (1993). Systems design analysis applied to launch vehicle configuration. NASA Technical Paper-3326.
• SIVARDIERE J. (1994). Kepler ellipse or Cassini oval. European Journal of Physics, Vol 15, pp.62-84. The New Encyclopedia Britannica (1987). Chicago, USA.
• WASHBURN A. (2010). A multistatic sonobuoy theory. Technical Report, NPS-OR-10-005, Naval Postgraduate School, Monterey, CA.
• WILLIS N.J. (2005). Bistatic radar. 2nd edition, SciTech Publishing.
• WILLIS N.J. (2008). Bistatic radar. Radar Handbook. 3rd edition, M.I. SKOLNIK (Editor in Chief), McGraw-Hill Professional, pp.23.4. Wolfram Mathworld (2012). Cassini Ovals. [Available at:] http://mathworld. wolfram.com /CassiniOvals.html. [Accessed: 10.06.2012]
• ZONG Y., YANG W., MA Q., XUE S. (2009). Cassini growth of population between two metropolitan cities - A case study of Beijing-Tianjin region. Chinese Geographical Science, Vol 19-3, pp.203-210.