The Product Of Weighted Lupa(？) Q-Bézier Curve And Its Application

The production, representation of the curves and surfaces are the main research con-tents of Computer Aided Geometric Design. In recent years, the representation of curves and surfaces are generalized by people from various perspectives. A new generalization of Bezier curves and surfaces with a q-integer has become a new hotspot in the research of parametric curves and surfaces. This thesis further made a generalization of weighted Lupa(？) q-Bezier curve and surface, based on the existing achievements of weighted Lu-pas q-Bezier curve and surface theory, combined with the product conclusions as well as various point projection algorithms of q-Bezier curves and surfaces, this paper from the following three aspects specific to research:weighted Lupa(？) q-Bezier curves produc-t, weighted Lupa(？) q-Bezier surfaces product and weighted Lupa(？) q-Bezier curves point projection. In this paper, the contributions include:First of all, in this paper we present the related conclusions about the weighted Lupa(？) q-Bezier curves product. The first conclusion is that the product of any two explicit weighted Lupa(？) q-Bezier curves is a explicit weighted Lupa(？) q-Bezier curve, it is based on the transformation matrix between basis functions and Then the second conclusion is obtained by reverse solving control polygon, that is any a explicit weighted Lupa(？) q-Bezier curve can be represented as a weighted Lupa(？) q-Bezier curve, and get the control vertices and weights of weighted Lupa(？) q-Bezier curve. By the above two conclusions can be deduced the following conclusion:the product of any two weighted Lupa(？) q-Bezier curves is a weighted Lupa(？) q-Bezier curve. At the same time, we give the corresponding numerical experiments to verify the correctness and feasibility of the product conclusions.Secondly, we move the results to the tensor product weighted Lupa(？) q-Bezier surface. The product of any two explicit tensor product weighted Lupa(？) q-Bezier surfaces is a ex-plicit tensor product weighted Lupa(？) q-Bezier surface. The product conclusion is deduced based on the change of bases. We get any an explicit tensor product weighted Lupa(？) q-Bezier surface can be represented as a tensor product weighted Lupa(？) q-Bezier surface, and get the control vertices and weights by reverse solving control polygon. Through the above two conclusions are easy to know the product of any tensor product weighted Lupa(？) q-Bezier surfaces can be expressed as a tensor product weighted Lupa(？) q-Bezier surface.Finally, we are combined with weighted Lupa(？) q-Bezier curves product to give the algorithm of weighted Lupa(？) q-Bezier curves point projection. When the shape parament value is one, the algorithm degenerates to the algorithm of general rational Bezier curves point projection, we get retrieval initial point by incremental algorithm, and constructs the geometric retrieval area. Then determines the retrieval end point by point location in geometric retrieval area, and cuts off the curve segment, which is located in the outside of the retrieval initial and retrieval end point. The geometric retrieval area becomes smaller and smaller during the subdivision process. A condition for terminating the subdivision process is provided, which is based on the squared distance function between the test point and the weighted Lupa(？) q-Bezier curve as well as weighted Lupa(？) q-Bezier curves product. Numerical experiments shows illustrate the accuracy and effectiveness of the algorithm.