الفهرس 
Abstract
Geometric preliminariesOnly 14 pages are availabe for public view
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Abstract
In this thesis, we study the evolution of a regular space curves, spherical images, Smarandache curves and surfaces generated by the evolution of these curves. Frames that associated to space curve are constructed and the derivative are obtained. Also, geometric visualization of these frames with Frenet frame are displayed. The model of the evolution of space curves depends on the frame that associated to the curve. Here we, used Frenet frame, Bishop frame and quasi frame to mobilize the curve in the space. Evolution of curves is represented by two sets of these frames. By applying compatibility condition on these frames and position vector of the curve, we obtained time evolution equation for the frames and curvatures. Exact solution of time evolution equation for curvatures are obtained. Curves corresponding to these curvatures are obtained via numerical integration of that frames. Finally, surfaces generated by evolving a regular space curve in R^3 as it evolves over time with arbitrary velocities attached to that frames are constructed. Geometric prosperities of these surfaces are discussed. We plotted these surfaces via numerical integration of Gauss-Weingarten equations.