Ayrık gruplar ve hiperbolik geometri
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Date
2008
Authors
Avcıoğlu, Osman
Journal Title
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Publisher
Uludağ Üniversitesi
Abstract
Bu çalışmada hiperbolik geometride konikler incelenmiş ve bunun için üst yarı düzlem modeli seçilmiştir. İnceleme iki bölümde gerçekleştirilmiştir.Birinci bölümde ikinci bölüm için hazırlık yapılmıştır.İkinci bölümde hiperbolik konikler incelenmiştir:İlk kısımda hiperbolik çemberin tanımı yapılmış, U da her hiperbolik çemberin bir Öklid çemberi, her Öklid çemberinin de bir hiperbolik çember olduğu ispatlanmıştır. İkinci kısımda hiperbolik elipsin ve yardımcı elemanlarının (odakları, merkezi, odak uzaklığı, asal ekseni, yedek ekseni) tanımı yapılmış, hiperbolik elipsin genel denklemi verilmiştir. Merkezi i olup odakları sanal eksen üzerinde bulunan hiperbolik elips (merkezil elips) incelenmiştir. Merkezil elips için elde edilen bulgular Möb(U) nun dönüşümleri kullanılarak U nun herhangi bir elipsine aktarılmış. Üçüncü kısımda hiperbolik hiperbolün ve yardımcı elemanlarının (odakları, merkezi, odak uzaklığı, asal ekseni) tanımı yapılmış, hiperbolik hiperbolün genel denklemi verilmiştir. Merkezi i olup odakları sanal eksen üzerinde bulunan hiperbolik hiperbol (merkezil hiperbol) incelenmiş, merkezil hiperbolün sonsuzdaki sınırını oluşturan noktalar ile asimptotları elde edilmiştir. Daha sonra merkezil hiperbol için elde edilen bu bulgular Möb(U) nun dönüşümleri kullanılarak U nun herhangi bir hiperbolüne aktarılmış, U nun herhangi bir hiperbolüyle ilgili istenilen tüm bilgilere ulaşılmış ve konuyla ilgili örnekler verilmiştir. Dördüncü kısımda hiperbolik parabolün ve yardımcı elemanlarının (odağı, doğrultmanı, ekseni, tepe noktası) tanımı yapılmış, hiperbolik parabolün genel denklemi verilmiştir. olmak üzere odağı , doğrultmanı i den sanal eksene dik olarak geçen hiperbolik doğru (ve böylece ekseni sanal eksen) olan hiperbolik parabol (merkezil parabol) incelenmiş, merkezil parabolün sonsuzdaki sınırını oluşturan noktalar elde edilmiştir. Sonra merkezil parabol için elde edilen bu bulgular Möb(U) nun dönüşümleri kullanılarak U nun herhangi bir parabolüne aktarılmış, U nun herhangi bir parabolü ile ilgili istenilen tüm bilgilere ulaşılmış ve konuyla ilgili örnekler verilmiştir.
In this study, the conics of hyperbolic geometry have been studied and for this study upper half plane has been used. The study consists of two chapters:The first chapter has basic studies for the second chapter.The second chapter has four sections studying conics of hyperbolic geometry:In the first section hyperbolic circle is defined and it is proved that every hyperbolic circle is an Euclidean circle and every Euclidean circle is a hyperbolic circle. In the second section hyperbolic ellipse, its focuses, center, focus distance, major axis and minor axis are defined and the general equation of an ellipse is given. The hyperbolic ellipse with center i, of which focuses are on the imaginary axis, the central ellipse, is examined. The findings obtained for central ellipse are transferred to an ordinary ellipse of U using the elements of Möb(U), so all required knowledge for an ordinary ellipse of U are obtained and related examples are given. In the third section hyperbolic hyperbola, its focuses, center, focus distance and major axis are defined and the general equation of an hyperbola is given. The hyperbolic hyperbola with center i, of which focuses are on the imaginary axis, the central hyperbola, is examined, the points of boundary at infinity and the asymptotes of the central hyperbola are obtained. After that, the findings obtained for the central hyperbola are transferred to an ordinary hyperbola of U using the elements of Möb(U). In the fourth section hyperbolic parabola, its focus, directrix, axis and vertex are defined and the general equation of a parabola is given. The hyperbolic parabola with the focus where and with the hyperbolic line passing through i perpendicular to the imaginary axis as the directrix (so with the imaginary axis as the axis), the central parabola, is examined and the points of boundary at infinity of the central parabola are obtained. After that, the findings obtained for the central parabola are transferred to an ordinary parabola of U using the elements of Möb(U), so all required knowledge for an ordinary parabola of U are obtained and related examples are given.
In this study, the conics of hyperbolic geometry have been studied and for this study upper half plane has been used. The study consists of two chapters:The first chapter has basic studies for the second chapter.The second chapter has four sections studying conics of hyperbolic geometry:In the first section hyperbolic circle is defined and it is proved that every hyperbolic circle is an Euclidean circle and every Euclidean circle is a hyperbolic circle. In the second section hyperbolic ellipse, its focuses, center, focus distance, major axis and minor axis are defined and the general equation of an ellipse is given. The hyperbolic ellipse with center i, of which focuses are on the imaginary axis, the central ellipse, is examined. The findings obtained for central ellipse are transferred to an ordinary ellipse of U using the elements of Möb(U), so all required knowledge for an ordinary ellipse of U are obtained and related examples are given. In the third section hyperbolic hyperbola, its focuses, center, focus distance and major axis are defined and the general equation of an hyperbola is given. The hyperbolic hyperbola with center i, of which focuses are on the imaginary axis, the central hyperbola, is examined, the points of boundary at infinity and the asymptotes of the central hyperbola are obtained. After that, the findings obtained for the central hyperbola are transferred to an ordinary hyperbola of U using the elements of Möb(U). In the fourth section hyperbolic parabola, its focus, directrix, axis and vertex are defined and the general equation of a parabola is given. The hyperbolic parabola with the focus where and with the hyperbolic line passing through i perpendicular to the imaginary axis as the directrix (so with the imaginary axis as the axis), the central parabola, is examined and the points of boundary at infinity of the central parabola are obtained. After that, the findings obtained for the central parabola are transferred to an ordinary parabola of U using the elements of Möb(U), so all required knowledge for an ordinary parabola of U are obtained and related examples are given.
Description
Keywords
Hiperbolik geometri, Hiperbolik metrik, Konikler, Hiperbolik konikler, Hyperbolic geometry, Hyperbolic metric, Conics, Hyperbolic conics
Citation
Avcıoğlu, O. (2008). Ayrık gruplar ve hiperbolik geometri. Yayınlanmamış yüksek lisans tezi. Uludağ Üniversitesi Fen Bilimleri Enstitüsü.