Study On Some Subdivision Schemes And Their Applications

The reconstruction is one of important questions for discussion incomputer-aided design. It is important that constructs subdivision scheme, theanalysis of convergence, the adjustment of the surface shape, and the applications ofcurves or surfaces modeling in the question of reconstruction. Studying curves orsurfaces modeling and their applications has an important significance to thedevelopment of computer-aided design.To solve inflexibility shortcomings of the classical four-point subdivisionscheme, this paper studies two classes of four-point subdivision scheme with threeparameters and gives the sufficient conditions of continuity for limit curves the firstscheme is C~0,C~1,C~2,C~3,C~4 and the second scheme is C~0,C~1,C~2) and surfaces(C~1).We can control the limit surfaces or curves through adjusting three parameters suitably.This method can model smooth approximating surfaces or curves numerically andtheoretically. The two schemes are characterized by simplicity and flexibility andoffer an effective tool for surfaces or curves modeling.We present four-point subdivision scheme with four parameters to enhancefigure抯controllability. We give sufficient conditions of continuity for limit curves (C~0to C~5) and surfaces (C~1) of four-point subdivision scheme with four parameters. Wecan control the limit surfaces or curves through adjusting four parameters suitably.This method can model smooth approximating surfaces or curves numerically andtheoretically and enrich surfaces or curves modeling. This paper gives geometricmeanings and functions to control figures of four parameters.We present hybrid subdivision scheme when initial mesh is hybrid polygon. Thealgorithm can be applied to triangular meshes, polygonal meshes and hybrid mesheswhich are open or closed topological structures. When the initial mesh is given, thestructure of the limit surface is concrete and can抰be adjusted. The phenomena ofshrink edge always occur when the topological structure is open domain in traditionalschemes. The main step is the operation of topological split firstly, then is the averageoperation of point geometry, and last is the revised operation of the vertex position. The shape control parameters are introduced in the operations of geometry averageand the revised vertex position. They can make the limit surface adjustable. This cansolve the two questions referred to above. We give and prove the sufficient conditionsof C~1 continuity of the limit surface.We propose surfaces modeling with adaptive refinements in order to solvecontradiction between efficiency and effect in constructing figures. The scheme hasmore flexibility than the traditional single subdivision scheme through adjustingcontrol genes and takes full advantage of the excellence of control Catmull-Calrksubdivision scheme and the excellence of uniform Catmull-Clark subdivision scheme.Surfaces are C~2 in regular mesh.We offer terrain simulation using four-point subdivision scheme with threeparameters and four-point subdivision scheme with four parameters respectively tosolve the problems of inadequate terrain simulation and large computation. When wemake the concave of the terrain simulation flat, we can utilize the former schemes, butthey make the holistic figure changeable a lot. When using our 4-point subdivisionscheme with three parameters and 4-point subdivision scheme with four parameterscarry out the terrain simulation, we can realize the adjustment of the concave throughrevising the parameters suitably and pertinently. They don抰 make the figurechangeable a lot. The two schemes can model ample terrain simulations rapidly andflexibility and solve the above two problems effectively through concrete samples.