Inverse Design Of Two Types Of The Surface With A Geodesic

With the beautiful intrinsic geometric features, the geodesic attracts widely concerns on both theory research and practical application. The inverse design of the surface through a geodesic has been one hot topic of CAGD subject in recent years. In this dissertation, both the design of the B-spline surface with a geodesic and the design of the approximate developable polyhedral surface with a geodesic are investigated, and the following two research results are obtained:Firstly, a design of parametric surface through the given non-uniform B-spline curve as geodesic is constructed. Furthermore, using random icity of the marching-scale functions, the antinomy between the irrational denominators of the Frenet frames and the rational denominators of the surface pencil’s NURBS expressions has been overcome neatly in the constitution of the surface pencil with a geodesic. Then, using the character of the discrete B-spline and new technique of the B-spline functions’products, the product of three marching-scale functions and the Frenet frames of the given curve has been precisely represented as the B-spline forms. And, preestablishing a series of weights in the pre-rational B-spline expressions for the marching-scale functions’factors, we improve the occasion, so that a group of family-parameters of the surface pencil can be obtained. Thus, finally, a design of non-uniform rational B-spline surface pencil through the given non-uniform B-spline curve as geodesic is constructed, and its explicit expression and algorithm are got. The formulas are pithy, the algorithm is simple and easy to realize. It is convenient for adjusting the shapes of the resulting surfaces, and we can keep the property that the given curve is a common geodesic of the resulting surfaces when their shapes are adjusted. The experiment shows that the algorithm is efficient and applicable.Secondly, as we all know, the mesh is a very popular type in modern industrial design and the developable surface or approximate developable surface is a widely applied surface. In this dissertation, the design of approximate developable mesh pencil with a common discrete geodesic is presented based on discrete Frenet frame of the given curve. This design overcomes the shortcomings of the conventional continuous surface design which restricts the continuity of the given curve and is inefficient. Further more, it is applicable in the two situations where all the given curves are continuous or discrete. The experiment results demonstrate that the algorithm is simple and efficient, being linear operation and stable, and also the constructed surfaces are approximately developable.