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Abstract
The development of objects employing ruled surfaces is of engineering importance because of the relative ease with which ruled surfaces can be manufactured without wasting the materials. The development using manual graphical methods of ruled surfaces is very simple but, the poses problem was in the error of the final solution as to its accuracy. The present work offers the convenient formulae describing the parametric equations of ruled surfaces and iLS classes, in which, two space curves (directrices), defining the edges of the surface are firstly introduced, then a set of rulings are constructed between the space curves. A study such as the conditions of develop ability, Gaussian curvature and geodesic curvature of the surface is essential job and requires a usual mathematical skill coupled with a deep understanding of differential geometry. The purpose of this thesis is to develop ruled surfaces using analytical methods. For that reason, an analytical algorithm is derived for developing a developable ruled surface into a plane and deriving the intrinsic equations of these developed surfaces. Some cases of developable surfaces are studied and developed. But during the application of this algorithm an arduous problem is found, that the integration of the intrinsic equations by elementary methods can not be practically obtained in a closed form. Then this thesis introduces a numerical method for integrating these intrinsic equations by using Rung-Kuna fourth order method which is solved by means of principles of mathematical programs, with maple9. Also, non-developable ruled surfaces are approximated and developed. Two methods are defined, the first method offers a direct algorithm to approximate and develop the surfaces by means of triangulation methods, and three examples are studied. The second method approximates the surface under the conditions of developability.