On the computational geometry;approximation of warped surfaces using developable surfaces

In this thesis, three approaches are derived to approximate ruled surfaces ,one approach has been introduced to developable ruled surfaces and the others are applied to non developable ruled surfaces. the first approach depends on the mapping of one directrix isometrically onto the plane by integrating a system of ordinary differential equations by analytical methods or numerical methods, the Sphericon, Oloid and the Helicoid are introduced as the applied examples of this approach. To over come the arduous and the difficulties of the analytical method a small modification is introduced depending on the conformal mapping ,by this way we have obtained the curvature at some specified points synthetically on the directrix curve of the ruled surfaces ,we offer the descriptive geometry of this way without expose to the analytical methods. On the other hand for the approximation of non developable ruled surfaces with a developable ruled surfaces , we have considered a relatively simple case of creating sequences of piecewise continuous developable surface patches using the Bilinear surface patch and connecting these patches together to achieve C1 continuity in order to produce a smooth and coherent surfaces .In the second approach the boundary curves of the ruled surfaces have been approximated using Lagrange curves or Bezier curves and we calculate the coordinates of the control points using the condition of the developability by solving a system of linear equations using single valued decomposition technique. The last approach is introduced for approximation by determining the torsal generators and torsal planes, then we cut the surface between two congruent planes after that we approximate the planar curves by Bezier or Lagrange curves at the end we obtain the bilinear surface patch.Future works1-When we have to connect the adjacent patches the boundaries of the these patches are sharpen lines is not smooth ,so we should consider continuity during computations.2-integerating the proposed algorithms to handle the ruled surfaces automatically and connect it with the Computer Aided Design (CAD).3-Within algorithmic Geometry (often Called ”Computational Geometry”) there is also a lack of results on basic algorithmic task in analytic and differential geometry.4-It is clear that the field of ”Computational Geometry” is a combination of classical ( projective, differential , analytic and descriptive) Geometry with new treatment by numerical method so, what will be the result when the ruled surfaces will be treated with finite element .