Optimized Design Of Surface Through The Geodesics

This paper studies the optimization designing of several classes of spline surfaces through the geodesics and the application in geodesic network interpolation. The types of spline are polynomial Bezier, polynomial B-spline and rational Bezier. And the geodesics include independent isoparametric geodesic and geodesic quadrilateral.In the case of polynomial Bezier surface. The bicubic polynomial Bezier surfaces are constructed to interpolating a pair or a set of independent isoparametric curves with degree 3 as the geodesics, and the control points of the interpolation surfaces are explicitly com-puted by the geodesic interpolation condition. The constraints for the quintic polynomial B6zier geodesic quadrilateral are discussed, and four quintic polynomial Bezier curvilinear quadrilateral with minimum strain energy is constructed to satisfy the constraints by a op-timized geometric method. Finally, we construct a polynomial Bezier surface with degree (7,7) to interpolate the quadrilateral as boundary geodesics by a optimized method, and the calculation of the control points are subregional.In the case of polynomial B-spline surface. For a set of independent isoparametric B-spline curves with degree 4, a B-spline surface with degree (5,3) is constructed to inter-polating two adjacent curves as boundary geodesics, and the adjacent surfaces are G1 con-tinuity along the common geodesic. The constraints for the polynomial B-spline geodesic quadrilateral are discussed, and four quartic polynomial B-spline curvilinear quadrilateral with minimum strain energy is constructed to satisfy the constraints by a optimized geo-metric method. Finally, we construct a biquintic polynomial B-spline surface to interpolate the quadrilateral as boundary geodesics. The control points are determined by subregional computingIn the case of rational Bezier surface. First, the constraints for the degree 4 ratio-nal Bezier geodesic quadrilateral are discussed, and four quartic rational Bezier curvilinear quadrilateral is constructed to satisfy the constraints by a optimized geometric method. Sec- ond, in order to compute linearly the control points related with the geodesic interpolation condition, we choose a degree (8,8) rational B6zier surface interpolating the quadrilateral as boundary geodesics. The control points and weights all are determined by subregional computing.In the case of the application in geodesic network interpolation. For the polynomial Bezier and B-spline geodesic network, we study the existence conditions of the surface to interpolating the two geodesic networks, and propose a method to Solve the corner coor-dination equations. When the corner coordination equations are satisfied, we can construct the surface independently to interpolate each quadrilateral as geodesic quadrilateral. And the surfaces which interpolate the geodesic networks can be obtained by combining all the surfaces. The surfaces are G1 continuity along the common geodesic.The innovation of this paper is mainly reflected in two aspects. First, a optimized ge-ometric method is proposed to construct the curvilinear quadrilateral which satisfies.the constraints of the geodesic quadrilateral. The algorithm is established by revealing the ge-ometric laws of the control points and weights. It can reduce the degree of the geodesic quadrilateral and make the curves with the global optimal characteristics. Second, the sur-faces are constructed by the subregional computing of the control points and weights, the control points related with the geodesic interpolation condition are explicitly computed by the geodesic interpolation condition, and the free control points and weights are determined by optimization technique of the surface energy. Compared with the traditional Hermite in-terpolation and Coons interpolation scheme, the methods proposed in this paper can reduce the degree of the interpolation surface, and the surfaces interpolate the geodesics with the global optimal characteristics. In the process of surface design, the surface adheres the NURBS standard and maintains the geometric intuition and excellent control properties. It satisfies the requirements of the curve/surface design in CAD system and has Potential application value in surface modeling.