Modeling With Geometric Constraints And Approximate Merging Of Curves And Surfaces

Curves and surfaces are common tools in CAGD systems, and the most operations in CAGD are based on Curves and surfaces. Both the shape modeling with geometric constraints which are from the given geometric information, and the approximate merging of curves or surfaces have become current research hotspots. In this dissertation, we have made deeply researches on the two topics and provided abundant and innovative results as follows:1. Curve modeling with points constraints means to get a curve to interpolate the given points. And the shape-preserving property is very important for the interpolation curves. In order to get the cubic uniformα-trigonometric/hyperbolic B spline curve, we blend a parametrized singular polyline and the trigonometric/hyperbolic polynomial B-spline curve using a blending factor, to automatically generate a C2 or G1 continuous hyperbolic polynomial B-spline with a shape parameter, which interpolates the given planar data points. By converting the first derivative or the curvature sign function of the interpolating curve into Bernstein polynomial, the nonnegativity conditions of Bernstein polynomial can be used to get the range of the shape parameter a and the necessary and sufficient conditions for the monotonicity or the convexity-preserving property of interpolation curves. The method is simple and convenient, need not to solve a system of equations or recur to a complicated iterative process.2. An algorithm for finding a G2 continuous, obstacle-avoiding curve in the plane is presented. Based on the guiding polyline path, both the rational quadratic parametric spline curve and the implicit functional splines curve are obtained. First, we partition the guiding polyline into control polygon sections by inserting several midpoints of polyline. Then, we find respective shape parameter of each curve section to avoid the vertices of the convex hull of an obstacle. At last, we choose the biggest shape parameter to avoid all the obstacles. Comparing with previous methods, the curves constructed by our approach have the following advantages:1. it is G2 continuous but with low degree; 2. it is shape-preserving, and the number of inflection point is the same as the one of the guiding polyline path; 3.it is obtained directly, and we need not to solve the fourth order equations; 4. the control polygon is visual, and we can adjust the curve easily. Specially, all shape parameters of cubic functional splines curve are local, so that they can be adjusted respectively with G2 continuity. Finally, several examples demonstrate the effectiveness and validity of the algorithm.3. A new algorithm for shape modification of triangular Bezier surface is presented by point constraints and normal vector constraints. Without any constraints or with the boundary continuity constraints at three corners, the new surface satisfies specified geometric constraints, such as multiple points and normal vector of the selected parametric point on the given triangular Bezier surface.With the help of Lagrange multipliers, the L2 norm between the surfaces before and after modification is minimized. The numerical examples show that this method is convenient for interactive design in CAD systems.4. Two kinds of approximate merging algorithms are presented:the approximate merging of two adjacent B-spline curves by one B-spline curve and the approximate merging of a pair of rational Bezier curves by interval Bezier curve. Applying the distance function between two B-spline curves with respect to L2 norm as the approximate error, we investigate the problem of approximate merging two adjacent B-spline curves into one B-spline curve. This method can be extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces very easily and successfully. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. So we can obtain both the new control points and the precise error of approximation explicitly in matrix form. On the other hand, based on the center curve and error curve of the interval Bezier curve, the approximate merging of a pair of rational Bezier curves by interval Bezier curve was obtained. The basic idea is to get the polynomial Bezier curve as the center curve by using the perturbation theory first. Then, we compute the error curve with constant or unconstant interval by solving linear equations or solving a quadratic programming problem. Both of the two error curves can be improved by the application of the well-known subdivision approach to this method. Furthermore, the interval Bezier curve can interpolate the rational curves at the two end points with the boundary constraints.5. Based on the orthonormality of triangular Jacobi polynomials and the transformation relationship between triangular Jacobi and Bernstein polynomials, the distance function between the original degree m triangular Bezier surface and the degree n(n≥m) approximate merging of 2 or 4 neighbouring triangular Bezier surface with respect to L2 norm was obtained. Without any constraints or with the boundary continuity constraints at three corners, we got the optimal approximate merging triangular Bezier surfaces by the least square method respectively, simultaneously the distance function reached the minimum. Both the control points of approximate merging surface and the precise error of approximation were expressed explicitly in matrix form. The algorithm is simple and direct, applicable for most cases. The degree elevation could reduce the merging error. Several numerical examples are presented to illustrate the correctness and validity of the algorithm.