Shape Adjustment And Analysis For Curves And Surfaces In CAGD

Shape adjustment and analysis for curves and surfaces are two inseparable research topics in computer aided geometric design(CAGD).On the one hand,shape adjustment requires profuse shape adjusting handles,which can help designers to modify shape intuitively and conveniently.On the other hand,shape analysis can both instruct designers accurately adjust shape and further manipulate geometric models.Our works are mainly on some problems in these two topics and our major contributions in this paper are as follows.On curves and surfaces with shape parameters.We study the relation between the known uniform B-spline basis functions with parameter and the classical B-spline basis functions.Based on the degree elevation of B-spline,we extend the uniform B-spline basis functions with parameter to ones with multiple parameters,which includes the known ones as special cases.Curves with shape parameters can easily be extended to tensor product surfaces by tensor product method.However,this method is not suitable for constructing non-tensor product surfaces with shape parameters.In this paper,we also present a set of quasi-Bernstein polynomials with one parameter,surfaces constructed by which are extension of Bezier surfaces over the triangular domain.Curves and surfaces with parameters presented in this paper have similar properties with uniform B-splines and triangular Bezier surfaces.Moreover,they are shape adjustable under the fixed control points.On adaptive shape adjustment for surfaces.We develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on the bivariate splines defined over Delaunay configurations.Our framework takes a triangulated set of points,and by solving a linear least squares problem and iteratively refining parameter domains with newly added knots,we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance.Unlike existing surface reconstruction methods based on triangular B-splines(or DMS splines),in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure.And shape of the fitting surface is adaptively adjusted.On the shape analysis of C-curves.As three control points are fixed and the fourth control point varies,the planar cubic C-curve may take on a loop,a cusp,or zero to two inflection points,depending on the position of the moving point.The plane can,therefore, be partitioned into regions labeled according to the characterization of the curve when the fourth point is in each region.This partitioned plane is called a characterization diagram. By moving one of the control points but fixing the rest,one can induce different characterization diagrams.We investigate the relation among all different characterization diagrams of cubic C-curves and conclude that,no matter what the C-curve type is or which control point varies,the characterization diagrams can be obtained by cutting a common 3D characterization space with a corresponding plane;The definition of Pythagorean-hodograph C-curve is given and a necessary and sufficient condition for a fourth order plane C-curve to be a PH C-curve is obtained.A new geometric construction approach of fourth order PH C-Bezier curves is also proposed.Finally,it is proved that higher order PH C-curves are either straight lines or polynomial PH curves.On the bound estimation of rational triangular Bezier surfaces.Based on the de Casteljau algorithm for triangular patches,also using some existing identities and elementary inequalities,we present two kinds of new magnitude upper bounds on the lower derivatives of rational triangular Bezier surfaces.The first one,which is obtained by exploiting the diameter of the convex hull of the control net,is always stronger than the known one in case of the first derivative.For the second derivative,the first kind is an improvement on the existing one when the ratio of the maximum weight to the minimum weight is greater than 2;the second kind is characterized as being represented by the maximum distance of adjacent control points.On the construction of Class A Bezier curves.The Class A Bezier curves presented by Farin were constructed by so-called Class A matrix,which are special matrices satisfying two appropriate conditions.The speciality of the Class A matrix causes the Class A Bezier to possess two properties,which are sufficient conditions for the proof of the curvature and torsion monotonicity.In this paper,we discover that,in[1],the conditions Class A matrix satisfied cannot guarantee one of the two properties of the Class A Bezier curves,then the proof of the curvature and torsion monotonicity becomes incomplete.Furthermore, we modified the conditions for the Class A matrices to complete the proof.