Researches On Offset Curve And Geodesic In CAGD

Offsets and geodesics are two fundamental curves with great importance in researches of computer aided geometric design(CAGD).The offset approximation in terms of rational representation,error estimation,and low-order approximation, as well as the surface design and modification passing through the given geodesic,has become one of the hotspots of investigation,due to the direct relationship with the function,quality,precision and efficiency of geometric design systems,engineering techniques,and processing industries.However,up to now there are still no satisfying solutions to the issues.Centering on those areas,this thesis carries out an in-depth study,and provides a series of convenient and efficient geometric algorithms.The abundant and innovative results are presented as follows:(1) Propose a high precision and high-order continuity offset approximation algorithm yielding precise circular offsetting.Based on the thorough analysis of all existing offset approximation methods,we abstract several principles for algorithm evaluation,including geometric interpretation,approximation precision, approximate curve representation and the approximation effect for curves in common use.According to these principles and based on the circle reparametrization theory,we apply a unique parametrization function,and deduce an innovative algorithm for planar offset approximation.The algorithm has the advantages of implementing precise offsetting for circulars without curve identification in advance,generating a much smaller number of curve segmentations and control points in the demand of high precision approximation,and improving the continuity of the resultant curve from G~1 to C~1.So it successfully overcomes different kinds of intrinsic disadvantages of existing methods,which cannot offset circulars precisely,or highly depend on the sampling technique,or cannot achieve global error control.The algorithm has practical significance in saving data storage,raising computing efficiency and improving the whole smoothness, so is especially suitable for geometric design systems. (2) Carry out a thorough analysis for error estimation of the offset approximation technique based on circle reparametrization.Focusing on the sourse of the approximation error,we discover and explain the duality of the two offset approximation methods based on circle reparametrization and circle approximation respectively.It reveals that the approximation error of the first method origins from the deflection of the approximate offset direction and the normal direction of the base curve,but the distance between the corresponding points of the approximate offset and the base curve is always equal to the given offset radius. In this sense,no precise offset approximation can be achieved until the thorough researches of the geometric meaning and the algebraic form of the deflection angle. We get rid of the error estimation way of measuring the distance between the corresponding points with the same parameter.By skillfully using the geometric information,Hausdorff distance estimation is proposed and successfully computed between the precise offset and the approximate offset.The discussion perfects the theoretical research on the circle reparametrization method, and provides theoretical basis for the comparison between the method and other algorithms.(3) Develop a new algorithm for low-order offset approximation in terms of fast implementation of curve modeling,and good compatibility of different geometric design systems.By improving the traditional vector Padéapproximation method and combining the curve subdivision and expansion technique at the mid-point,the method yielding arbitrary order rational offset approximation and achieving the given precision is obtained.The method avoids the limits of the sampling technique,which leads to unstable results and cannot achieve global error control.Instead it applies the linear system for the final solution,so the computation and error estimation is convenient and fast in implementation.The method provides a new geometric tool for low-order offset approximation.(4) Provide a technique for designing and modeling surfaces with the given curve as a geodesic.The previous research only proposed a common principle without the consideration of the surface representation and modification requirements in geometric design systems.Our method solves the problem,developing a new way for design automation in garment manufacture and shoe-making indus- try.Considering the demand of data representation and exchange in computer aided design and computer aided manufacture(CAD/CAM) system,we research the rational surface design,and provide an algorithm for the construction of the cubic rational surface.Variation optimization technique is used,which can be operated directly on the surface in an intuitive way,giving consideration of both surface modeling and smoothing.(5) Offer a wholly bran-new idea for developable surface design.Based on the view of regarding a surface as a locus of the moving point of the given curve moving along the local frame in the space,we deduce the sufficient and necessary conditions for the surface passing through the given curve to be developable. Inspired by the practice in garment manufacture and shoe-making industry,we give special attention to the situation when the given curve is a geodesic on the surface.According to the types of developable surfaces,the given curve is characterized in order to make the algorithm more convenient and effective in engineering application.Besides,rational representation is provided,which further improves the computation efficiency.(6) Discuss the singularity distribution of H-Bézier curves,which can precisely represent the widely used curves,hyperbola and catenary.Using the moving control point technique,the correspondence between the distribution of the moving control point and the curve singularities is characterized.Comparison in terms of the type of the discriminant curve is provided between the cubic H-Bézier curve,Bézier curve,rational Bézier curve and C-Bézier curve.We also illustrate the application of this curve characterizing method.