Dual uzayda çatı hareketleri ve ardışık teğet kongrüanslar /
Yazar: EMİN ÖZYILMAZ
Danışman: PROF.DR. ALİ ÇALIŞKAN
Ege Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı
Derece: Doktora, 1996
Konu: Matematik = Mathematics
Dizin: Geometri = Geometry ; Kongrüans = Congruence ; Uzay = Space ; Çatı = Roof

Özet:
ÖZET Üç bölümden oluşan bu tezin birinci bölümü, daha sonraki bölümlere temel teşkil eden bilgilere ayrılmıştır. ikinci bölümün birinci kısmında bir yüzeyin herhangi bir P noktasından geçen bir (a) eğrisi ile yine aynı noktadan geçen (ctı) ve (a2) parametre eğrilerinin bu noktadaki Darboux çatılarının Darboux vektörleri arasındaki bağıntı [1], Serret-Frenet çatılan için verilmiştir. Böylece, Frenet vektörleri arasında kurulan bu bağıntı yardımıyla Diferansiyel Geometrinin bazı klasik formüllerine ulaşılmıştır. İkinci bölümün ikinci kısmında bir doğru kongrûansına ait herhangi bir (A,) regle yüzeyi ile A0ortak doğrusuna sahip (An) ve (A^) parametre regle yüzeylerinin A0daki Blaschke çatılarının çeşitli hareketlerine ait dual Stainer vektörleri arasındaki bağıntılar ortaya konmuştur. İkinci bölümün son kısmında bir doğru kongrûansı içindeki regle yüzeylerin Blaschke çatılarının çeşitli hareketlere ait adımları ve aralarındaki bağıntılar verilmiştir. Son bölümün ilk kısmı [5] de tanımlanmış olan bir doğru kongrüansımn A0 doğrusundaki teğet kongrüans kavramının ardışık teğet kongrüans kavramına genelleştirilmesine ayrılmıştır. Bu genelleştirmede [A] kongrûansı ile [A] kongrüansımn A0 doğrusuna bağlı [Y] teğet kongrûansı arasındaki formüller, ardışık teğet kongrüanslar için yazılarak bu kongrüanslara ait büyüklüklerin [A] kongrüansımn A0 doğrusuna ait büyüklükleriyle irtibatı sağlanmıştır. Sonra, sırayla, [A] ve [Y] teğet kongrüanslannın A0 ortak doğrusuna sahip (Aj) ve (Y:) regle yüzeylerinin dual küresel eğrilikleri arasındaki bağıntı verilmiş, daha sonra [Y] teğet kongrüansımn teğet dikkonoitlerine ait dual eğriliği, drali ile (A:) regle yüzeyinin AQ doğrusundaki dual eğriliği ve drali arasındaki ilişkiler ortaya konmuştur. Ayrıca [Y] teğet kongrüansımn bir (Yj) regle yüzeyi, (Yn)ve (Y^) parametre regle yüzeylerinin bir Y0 ortak doğrusundaki Blaschke çatılarının Blaschke vektörleri arasındaki bağıntı bulunup, çeşitli özel halleri incelenmiştir. Bundan başka, (Yj), (Yn) ve (Ya) regle yüzeylerine 58relation between dual curvature and dralls belonging to tangent right conoid of tangent congruence [Y] and the line A« of the ruled surface (Aj) has been showed. Moreover, the relation among the Blaschke vectors of Blaschke trihedrons at a common line Yo of to any ruled surface (Yi) and parameter ruled surface (Yn), (Y21) of the tangent congruence [Y] has been found and several special case of this relation has been examined. Besides this, the relations have been expressed between some magnitudes (dual arc element,dual curvature) of the ruled surface (Yi), (Yn) and (Y2i) and belonging to the line Ao of the ruled surfaces (Ai), (An) and (A2i) of the congruence A(u,v). In the second part of the third chapter, a new congruence [Ax] has been defined as putting a tangent right conoid on each line of a ruled surface (Ai) of a line congruence [A]. Thus, inspiring by [16], a new dual trihedron which is not the Blaschke trihedron whose geometric interpretation is more determinable, at a striction point of line Ao of the ruled surface (At) has been constructed. The Blaschke and the new dual trihedron has been examined like Serret-Frenet and Darboux trihedrons defined on a curve on surface. In the third part of the third chapter, by considering special case of the congruence [Ax] which has been defined in the previous part, the concept of tangent congruence, tangent right conoid, drall and the relation among Blaschke vectors of Blaschke trihedrons of ruled surface, having common line Ao, has been examined for this special case. At the end of this chapter, the concept of tangent congruence for the some special congruences has been examined. 61

Abstract:
SUMMARY The first chapter of this thesis which is composed of three main chapters, is reserved to informations which are the bases for the latter sections. In the first part of the second chapter, the relation, between reel Darboux (instantaneous) vectors of any curve (a) which passes through any point P of a surface, and parameter curves (ctı), (a2) passing through the same point given in [1], are given for Serret-Frenet vectors. Then, by this relation established for Frenet vectors, we have some classic formulas of the Differantional Geometry. In the second part of the second chapter, the relations among dual Steiner vectors belonging to several motions of Blaschke trihedrons at the same line Ao of a ruled surface (Aj) and parameter ruled surface (An), (A2O which has same line Ao of a line congruence A(u,v)., have been put forward. At the end of second chapter, it has been given pitch belonging to several motions of Blaschke trihedron of ruled surface of a line congruence and relations between them. The first part of the last chapter is about the generalization of the concept of tangent congruence at the line Ao of a line congruence defined in [5] to consecutive tangent congruence concept In this generalization, the formulas between congruence [A] and the [Y] tangent congruence of [A] connected to line Ao are written for consecutive tangent congruence and the relations between elements belonging to these congruences and the elements belonging to line As of congruence [A] is constructed. Then, the relation among dual spherical curvatures of ruled surfaces (Ai) and (Yj) which have common line Ao of a line congruence A(u,v) and Y(u,v), have been given. After that, 60relation between dual curvature and dralls belonging to tangent right conoid of tangent congruence [Y] and the line A« of the ruled surface (Aj) has been showed. Moreover, the relation among the Blaschke vectors of Blaschke trihedrons at a common line Yo of to any ruled surface (Yi) and parameter ruled surface (Yn), (Y21) of the tangent congruence [Y] has been found and several special case of this relation has been examined. Besides this, the relations have been expressed between some magnitudes (dual arc element,dual curvature) of the ruled surface (Yi), (Yn) and (Y2i) and belonging to the line Ao of the ruled surfaces (Ai), (An) and (A2i) of the congruence A(u,v). In the second part of the third chapter, a new congruence [Ax] has been defined as putting a tangent right conoid on each line of a ruled surface (Ai) of a line congruence [A]. Thus, inspiring by [16], a new dual trihedron which is not the Blaschke trihedron whose geometric interpretation is more determinable, at a striction point of line Ao of the ruled surface (At) has been constructed. The Blaschke and the new dual trihedron has been examined like Serret-Frenet and Darboux trihedrons defined on a curve on surface. In the third part of the third chapter, by considering special case of the congruence [Ax] which has been defined in the previous part, the concept of tangent congruence, tangent right conoid, drall and the relation among Blaschke vectors of Blaschke trihedrons of ruled surface, having common line Ao, has been examined for this special case. At the end of this chapter, the concept of tangent congruence for the some special congruences has been examined. 61