الفهرس يوجد فقط 14 صفحة متاحة للعرض العام
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المستخلص
surface design and its applications are widely used in CAGD (Computer Aided Geometric Design) and industrial areas. The cylinder and the airfoil are ruled surfaces with particular shapes. In this thesis, dual space drawing methods for these particular ruled sUrfaces are proposed. The underlying principles of these methods are the applications, extensions and variations of the dual (space) de Casteljau algorithm .FIrSt some surfaces are created by using the initial dual de Casteljau algorithm directly. The cylinder and the airfoil, on the other hand, are created by using the extensions or variations of the dual de Casteljau algorithm. During the drawing processes, the screen representation of the rulings is presented. This treatment presents the rulings as clipped line segments on the computer screen in order to show the proportion of the ruled surface on the computer screen. In addition, the dual characteristics of Bezier surfaces are also discussed. The special construction method of the displacement matrix with two parallel control screws is demonstrated. The normal screw is normalized into the unit screw as well. These new methods show the universal possibility of the dual space approach for drawing ruled surfaces. All the work presented is based on lie group theory.
The thesis is made up of seven chapters, which are.
Chapter (1)
It is an introductory chapter concerning computer graphics and its subfields like geometry, animation and rendering (scattering and transport).Also we give a brief history of computer graphics .then we talk about kinematic generation of ruled surface including the meaning of ruled surface .Some examples of ruled surface are given .Also some applications of ruled surfaces are mentioned and a brief history of ruled surface generation .At the end of the chapter we mention the contribution of the work.
Chapter (2)
In this chapter we explain all backgrounds which are needed for the work. Ftrst, we give an introduction to group theory and other differential geometry definitions like Differentiable manifolds, differentiable maps, and Tangent spaces. We give a lot of illustrative examples of manifolds. Then, we explain the concept of lie group theory which is based on algebraic concept of a group, and the differential-geometric notation of a manifold .We give a brief history of lie Group, concept of lie group and some examples of lie group.
As we mention the relation between any manifold and its tangent space there is a relation between lie group and its lie algebra which also explained in 1his chapter .Also the exponential map which maps from lie Algebra to its corresponding lie group is mentioned.
Through this chapter, we define a very important number system which is caIIed the dual number system, through all the work we use dual number representation of a line which is called Pliicker coordinates. In this chapter, we explain this representation of line and why displacement of line form a lie group and how we shall benefit from lie group concept and how we get lie algebra (tangent space at identity) by dual space algorithm and then get the corresponding Lie group (displacement of line) by exponential map. Ftnally, we explain ruled surface as curves on the Unit Sphere.
Chapter (3)
This chapter contains a survey of two other previous methods generating ruled surfaces the fIrst was published in 1997 and the second was published in 2002.
Chapter (4)
This chapter is concerned with the theory and algorithm that all the work based on.
First. we explain screw theory in details including defInition of screw, a screw as a dual 3¬vector and the displacement of a screw. Then, we explain the initial real De Casteljau algorithm which generates an interpolated Bezier curve using n+ 1 control points, the properties of Bezier curve mentioned .Then, we make a great map from Real Algorithm to dual algorithm replacing control points by control lines which generate interpolated Bezier surface instead of Bezier curve.
Chapter (5)
All the work in this chapter and all theoretical methods have been confIrmed using the C programming language under Windows. In this chapter we explain how we draw particular geometrical shape ruled surfaces using the Variation of the Dual De Casteljau Algorithm and what are the conditions required to draw a ruled surface directly from dual algorithm and why we cannot do that in case of the cylinder and the airfoil surface and what are the boundary conditions we need to add to generate those surfaces. Then, an important issue explained, why the interpolation line is the tangent vector of the Bezier Curve. Finally we show the graphical results of computer program which are drawn by entering the numerical results of the program to the AutoCAD application.
Chapter (6)
This chapter is concerned with computer implementation issues .An important problem which faced us was how to draw rulings (lines) which represented by Six Pliicker coordinates on the screen without any need to transform back from Plucker to 3D points, this is what we explain in this chapter. Finally the code structure explained and some numerical results.
Chapter (7)
This chapter contains two comparisons between our method and two other methods.
Then the conclusion about the advantages and disadvantages of our method is presented.
Chapter (8)
This chapter contains the conclusion and future work.
Appendix
Contains the C program code