Geometric Algorithms Of Toric Surfaces And Toric Volumes

Parametric curves and surface patches,which include Bezier curves/surfaces,B-spline curves/surfaces and NURBS curves/surfaces,etc,play an important role in computer aided geometric design(CAGD)and computer aided design(CAD).De Casteljau algorithm and degree elevation are two important geometric algorithms in the modeling method of curves/surfaces and frequently used in geometric design of subdivision,composite curves and sweeping surfaces.In CAGD,parametric surfaces are commonly used to represent the boundaries of objects in threedimensional space.With the development of modern industries such as aircraft and automobile,solid models of complex structures used in industry need not only boundary definitions but also internal definitions,so it is very important to study the representation and properties of 3D parametric volumes.Toric surface patch is a generalized form of classical rational Bezier surface.It is a kind of multi-sided rational parametric surface,which preserves many excellent properties of rational Bezier surfaces.In this thesis,the depth of Toric surface patches is used to replace the order of rational Bezier surface,we present the display formula of the de Casteljau algorithm and degree elevation of Toric surface patches.The degree elevation algorithm of Toric surface makes it possible to use the Toric-Bernstein basis function for the refinement process of Isogeometric analysis(IGA),which is similar to the p-thinning given by the degree elevation based on NURBS.Finally,We also apply the degree elevation of Toric surface patches to IGA,for the polygon physical domain of the parameterized by two-dimensional Toric surface,using the degree elevation algorithm to refine the physical domain,can effectively improve the accuracy of the solution.The 3D parametric volume model is widely used in engineering applications.In this paper,we study the geometric algorithms of Toric body volumes.First,using the related theories of Minkowski sum of the 3D lattice point set and discrete convolution of the function sequence,we give the recursion and degree formula of the Toric-Bernstein basis functions of three variables,and then give the de Casteljau algorithm and degree elevation algorithm from d to(d+1)of Toric volumes,extend the degree elevation algorithm to any number of degree and apply preliminarily the degree elevation of Toric heptahedron to IGA.At the same time,a preliminary discussion was made on the approximation properties of the control mesh of the Toric surfaces/volumes during the degree elevation process.The key to the construction of parametric curves/surfaces is the construction of corresponding basis functions.The properties of parametric curves/surfaces such as Bezier curves/surfaces,NURBS curves/surfaces can be directly obtained from the properties of their corresponding basis functions.In this thesis,we define a new kind of blending functions associated with a real points set,called generalized Toric-Bernstein(GT-Bernstein)basis functions.Then the generalized Toric-Bezier(GT-Bezier)curves and surfaces are constructed based on the GT-Bernstein basis functions,which are the projections of the(irrational)Toric varieties in fact and the generalizations of the classical rational Bezier curves and surfaces and Toric surface patches.Furthermore,we also study the properties of the presented curves and surfaces,such as coner points interpolation property,multiple knots property and Toric degeneration property.