Curves And Surfaces With Shape Parameters And Their Applications In Progressive Iterative Approximation

In recent years, curves and surfaces with one shape parameter or multiple shapeparameters have drawn many scholars’ attention for their flexibility, simple formulationand so on. The main idea is that, the curves/surfaces can be adjusted by changing the shapeparameters without moving their control points/control meshes. Thus, we can realize theentire or local adjustment of the curves/surfaces and change the distance between thecurves/surfaces and its control points/control meshes. What’s more, as Progressive IterativeApproximation is a new kind of fitting and approximation technique, its intuitivemathematical idea is that, the limitation of the curves or the surfaces that generated by thePIA method can fit the initial data points. It is already proved that the Normal TotallyPositive bases satisfy this property. As the PIA method can be widely used in thereconstruction of curves or surfaces, and the data structure design of free curve and surfacereconstruction is one of the key technologies of reverse engineering, we do the followingwork.Firstly, a class of exponential uniform B-spline curves and surfaces with multipleshape parameters is provided. They preserve the main properties of the exponential uniformB-spline curves and surfaces, such as positive property, convexity-preserving property andso on.Secondly, the curves can be adjusted their shapes by changing the values of shapeparameters without moving their control points. So we can realize the entire or localadjustments of the curves.Thirdly, the exponential uniform B-spline curves can represent hyperbola, catenaryand other transcendental curves precisely. The corresponding surfaces can be constructedby using tensor product, so their properties are similar to that of the curves.Fourthly, geometric meaning of the shape parameter is analyzed, and we give out howthe shape parameter adjusts the curves and the surfaces.Fifthly, and we study the PIA property of curves/surfaces with one shape parameter,we also prove that we can improve the PIA convergence rate by changing the shapeparameter. So, have a completely different way to improve PIA convergence rate.At last, due to the particular advantages of triangular surfaces in CAD, we extend thePIA property to the bivariate Bernstein bases with one shape parameter over a triangledomain. We also prove that we can improve the PIA convergence rate by changing theshape parameter.