Approximating Curves And Surfaces With Constraints

Bézier curves and surfaces are widely used modelling tools in CAD/CAM systems.Degree reduction of parametric curves and surfaces,which tries to approximate a given curve or surface of certain degree by another one of lower degree,has become an important and hot problem in CAGD.It has many applications in geometric modelling,such as data exchange,data compression and data comparison.In this thesis,we have made a systemic theoretic research on the problem of degree reduction of Bézier curves and triangular Bézier surfaces in CAGD.The main creative results are as follows.Firstly,in contrast to traditional methods,which typically consider the components of the curve separately,we use geometric information on the curve to generate the degree reduction.And the constraint of G²-continuity is introduced for the approximation problem,so positions,tangents and curvatures are preserved at the two endpoints.We then present a novel approach to consider the multi-degree reduction of Bézier curves in L₂-norm.The optimal approximation is obtained by minimizing the objective function based on the L₂-error between the two curves.In order to avoid the singularities at the endpoints,regularization terms are added to the objective function.Furthermore,for the special case of G¹-continuity,we presents another approach to solve the approximation problem in terms of the quadratic programming method,with linear constraints to satisfy the coincidence of tangents at the endpoints.Then,degree reduction is changed to solve a quadratic problem of two parameters with linear constraints.We apply the new approach to improve the parameterizations of approximating curves to be close to arc-length parameterizations.Secondly,a polynomial curve on[0,1]can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind.We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other,and examine the stability of this linear map.In the p=1 and∞norms, the condition number of the Chebyshev-Bernstein transformation matrix grows at a significantly slower rate with n than in the power-Bernstein case,and the rate is very close(somewhat faster) to the Legendre-Bernstein case.Using the transformation matrices,we present a method for the best multi-degree reduction with respect to the(t-t²)^1,2-weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multidegree reduction with constraints of endpoints continuity of orders r,s(r,s≥0). This method has a quadratic complexity,and may be ill-conditioned when it is applied to the curves of high degree.We estimate the posterior L₁-error bounds for degree reduction.Finally,for a given triangular Bézier surface of degree n,we investigate the problem of approximating it by a triangular Bézier surface of degree m with boundary constraints.We constrain continuity conditions at the three corners of triangular Bézier surfaces,so that the boundary curves preserve endpoints continuity of any orderα.The l₂ and L₂ distances combined with the constrained least-squares method are used to get the matrix representations for the control points of the degree reduced surfaces.Both methods can be applied to piecewise continuous triangular patches or to only a triangular patch with the combination of surface subdivision,the resulting piecewise approximating patches are globally C⁰ continuous.Also,we estimate the error of approximation and provide some examples to demonstrate the effectiveness of the two methods.