Mannheim offsets of ruled surfaces in real and dual spaces

This thesis consists of seven sections.In the first section, motions in 3-dimensional Euclidean space , ruled surfaces and striction line and distribution parameter of these surfaces are given. Also, Frenet frames and Frenet invariants of the ruled surfaces are expressed clearly. Moreover, closed ruled surfaces and their integral invariants are brought to mind.In the second section, we talk about the space of dual vectors and E. Study mapping, ruled surfaces in dual space and their Blashcke frames. The geometric interpretation of dral and dual angle of pitch of closed ruled surfaces are introduced.In the third section, Mannheim offsets of the ruled surfaces are investigated in real and dual spaces. First, the theorems and corollaries given by Orbay and et al are given briefly. Then, by considering closed ruled surfaces, Mannheim offsets of ruled surfaces are studied. The relationships between a ruled surface and its Mannheim offset surface are given.Section four consists of a summary of Minkowski 3-space, timelike and spacelike ruled surfaces and their Frenet formulas and Frenet invariants.In the fifth section, ruled surfaces are considered in dual Lorentzian space. By considering E. Study mapping, geometry of dual hyperbolic and dual Lorentzian spherical curves are investigated.In the sixth section, Mannheim offsets of ruled surfaces are considered in Minkowski 3-space. According to the classifications of timelike and spacelike ruled surfaces, Mannheim offsets of these surfaces are defined and some theorems and corollaries related to developable offset surfaces are obtained.In the seventh section, the last section, closed ruled surfaces are considered in dual Lorentzian space. Mannheim offsets of these surfaces are investigated and relationships between real and dual invariants of offset surfaces are given.The original sections of this thesis are second part of the third section in which the Mannheim offsets are studied in dual space and all of the Section 6 and Section 7.