الفهرس 
2 Quasi Frame and Equations on Smarandach Curvesيوجد فقط 14 صفحة متاحة للعرض العام
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المستخلص
Abstract
In this thesis, we investigate the quasi frame in four-dimensional Euclidean space E4, three and four-dimensional Minkowski space and , and apply the results of the quasi frame in four-dimensional Euclidean space E4 on different types of Smarandach curves. Also, we show that the quasi frame is an alternative frame of the Frenet and Bishop frame, but easier in computations, has no singularities, has more accuracy, and is considered as a generalized frame of both Frenet and Bishop frames. Furthermore, we introduce some fundamental mechanical elements from the perspective of differential geometry to show another side of visualization to understand what these elements mean depending on the Frenet formulas that describe the kinematic properties of a particle moving along a continuous, differentiable curve in 3-dimensional Euclidean space E3, we study the geometric properties of the curve itself irrespective of any motion. Thus, we shed the light on snap or jounce which is one of the most important topics in mechanics. We study the snap vector in the planar and space motion, for planar motion, we take the oscillation of simple pendulum, central force proportional to distance, and Keplerian orbital motion as models to understand the meaning of this concept, for space motion, we take the electron moving along a right-handed circular motion under constant magnetic field and a particle moving along a logarithmic spiral spring as models to ensure the physical meaning that we supposed. Then, we discuss the geometric properties of the quasi-Hasimoto surfaces in for the three cases of non-lightlike curves and obtain the Gaussian and mean curvatures of each case. Then, we give a necessary and sufficient condition of the quasi-Hasimoto surfaces in to be developable surfaces. Also, we give a characterization of parameter curves of quasi-Hasimoto surfaces in . Thus, we give necessary and sufficient conditions of the s-parameter and t-parameter curves of quasi-Hasimoto surfaces in to be geodesics, asymptotic and lines of curvature. Finally, applications on quasi curves and their correspondence quasi-Hasimoto surfaces are given.