Research On Subdivision,Merging And Extension Of Quartic Q-ball Curve And Surface

In the field of CAD/CAM,parametric curves and surfaces are the primary tool for describing geometry shapes of product,and are also the foundation and core of CAGD.The construction of curves and surfaces with good properties is of great value in theory and application.The ascending and descending order calculation speed of Ball curves and surfaces is better than Bezier curves and surfaces.Moreover,the degenerate condition is simpler than the Bezier method,but the Ball method can not adjust its shape after given control points.In recent years,generalized Ball curves and surfaces with shape parameters have become a new research hot-spot in the field of CAGD,these new curves and surfaces inherit the advantages of traditional Ball curves and surfaces,and contain independent shape parameters,so it is convenient to adjust and control their own shapes.Based on this background,this paper mainly have done a deep research on the subdivision,approximate merging and extension algorithms of a class of quartic Q-Ball curves and surfaces.The main contents and achievements are as follows:(1)Firstly,the coefficient subdivision method of quartic Q-Ball curve is proposed.By comparing the power term coefficients in the curve expression before and after subdivision,the subdivision condition of the curve and the expression of the curve control points after subdivision are obtained,thus avoiding the curve derivation process.Secondly,a tangent subdivision method for quartic Q-Ball curves is proposed.The method keeps the tangent lines of the curves before and after the subdivision are same in dividing point,and then adopt the de Casteljau fixed ratio subdivision idea,the expression of the control points of each curve segment after subdivision is obtained.Finally,the idea of curve subdivision algorithm is applied to the surface,and the coefficient subdivision algorithm and tangent plane subdivision algorithm of quartic Q-Ball surface are proposed respectively.Numerical examples show that the proposed methods can achieve arbitrary subdivision and local shape adjustment of curves and surfaces.(2)In order to reduce the amount of data transmission in CAD/CAM system,an approximate merging method of 2-adjacent quartic Q-Ball curves based on GIMT and parameterization of arc length is proposed.This article adopts arc length parameterization is a decomposition method with equal arc length.Because the classical dichotomy principle is used in the parameterization process,the problem caused by non-uniform arc length parameterization is effectively solved.Finally,the generalized inverse matrix theory is used to solve the display expressions of control points of merged curves,and an approximate merging with constraints of endpoints interpolation and without constraints of endpoints interpolation are made for the merging process.The experimental results show that the proposed method has obvious effect on curve merging and greatly improves the merging error and curve approximation of two adjacent quartic Q-Ball curves.(3)Because the representation ability of a single quartic Q-Ball curve is limited,this paper derives the extension algorithm of the quartic Q-Ball curve,which satisfying C2 and G2 continuous to fixed points,and C1 and G1 continuous to fixed curves and through two points.The grey wolf optimizer is a new intelligent algorithm,which can intelligently optimize the approximate expression of the physical deformation energy of the continuation curve.The optimized continuation curve is smoother and the energy is minimum.Simulation graphics show that the curve has a very good continuation effect and the algorithm has high practicability.It can be used to express and improve the modeling of some more complex curves.