The Regular Control Surfaces Of Toric Patch

NURBS(Non-Uniform Rational B-Splines)curves/surfaces are widely used in curve and surface design.They play an important role in Computational Geometry,Computer Aided Geo-metric Design(CAGD),Computer Aided Design(CAD),Geometric Modeling and some related fields.The classical rational Bezier curves/surfaces and B-spline curves/surfaces are special cases of NURBS curves/surfaces.The modification of weights is one of methods to control the shape of curves/surfaces.Especially,the extreme weight pulls the NURBS curve/surface to the corresponding control point,which is well known the geometric meaning of a single weight of NURBS curves/surfaces.Toric patch is a kind of multisided rational parametric surface,which is associated to a given finite integer lattice points set A.The theory of toric patch is derived from algebraic geometry and combinatorics,and its basis functions are the generalization of Bernstein basis.The classical rational Bezier curves/surfaces are special cases of toric patches.At present,geometric property of toric patches is also one of research hot spots.Garcia-Puente,Sottile and Zhu pointed out when all weights tend to infinity,the limiting of toric patch is its regular control surface.As the generalization of this conclusion,when all weights tend to infinity,Zhang and Zhu presented that NURBS curve/surface tends to its corresponding regular control surface.Regular control surface is a kind of piecewise C0 surface,which is determined by regular decomposition of finite integer lattice points set A.Thus,the different regular decompositions induce different regular control surfaces.Then a natural question is how many regular control surfaces of a toric patch?And,how many regular control curves/surfaces of a NURBS curve/surface?In this thesis,firstly,based on the theory of secondary polytope in combinatorics,we present an algorithm to compute the number of regular control surfaces of a toric patch.Secondly,the construction of the regular control surfaces and another algorithm are given by using the state polytope in combinatorics and integer programming.Finally,we extend these results to study the regular control curves/surfaces of NURBS curve/surface.Together with toric degeneration of NURBS curves/surfaces,we give the upper bound of the number of regular control curves of NURBS curve and the upper bound of the number of regular control surfaces of a class of special biquadratic NURBS surface.In Chapter 1,we recall the history and the backgrounds of parametric curves and surfaces briefly,including rational Bezier curves/surfaces and NURBS curves/surfaces.We illustrate the backgrounds and developments of toric patches,and its application in geometric modeling.In Chapter 2,we introduce the definition and properties of toric patch and NURBS curves/surfaces.We recall the toric degeneration of toric patches,which explains the meaning of the limiting surface of toric patch when its weights tend to infinity.Chapter 3 studies the number of regular decompositions of finite integer lattice points set and presents the relationship between the regular decompositions and the secondary polytope.Furthermore,we indicate that the number of regular control surfaces of toric patch associated with finite integer lattice points set is equal to the number of regular decompositions of finite integer lattice points set.We also present an algorithm to compute the number of regular control surfaces of toric patch.In Chapter 4,for finite integer lattice points set,we consider the construc-tion and the number of regular decomposition by using the method of the integer programming and point out the relationship with the state polytope.At last,we can obtain the construction and the number of the regular control surfaces.Another algorithm to compute the number is also provided.In Chapter 5,based on toric degeneration of NURBS curves/surfaces,we study the number of regular control curves/surfaces of NURBS curve/surface,and the upper bound of the number is presented.Chapter 6 concludes the whole thesis and presents the future work.