Interproximate curve subdivision
Subdivision is a process that for a given coarse polygon (control polygon) recursively refines it using some predefined rules. This is a very popular method of generating curves. The subdivisions schemes can be divided in general into two categories: approximate and interpolatory. In the paper, a subdivision scheme which combines subdivision schemes of both categories is presented. This type of subdivision is called an interproximate subdivision scheme. The authors take the 4-point subdivision scheme as the interpolatory and the cubic B-spline as the approximate scheme. In the proposed scheme, they can choose which vertices of the coarse polygon should be interpolated and which of them should be approximated. Moreover, the authors present two extensions of the proposed scheme. The first extension makes the scheme convexity preserving and the second relaxes some of the vertices from the interpolation requirement during the iteration process. In the beginning, the authors briefly present the 4-point and cubic B-spline subdivision schemes. Next, they introduce the geometric rules for the interproximate scheme. The vertices are divided into two categories: interpolatory (\(I\)) and approximate (\(A\)). To generate an edge point between two \(I\) vertices they use the 4-point scheme, and between two \(A\) vertices they use the cubic B-splain scheme. In the case when the one end of the edge is \(I\) and the other \(A\), they derive a formula for the new vertex, which is a combination of both schemes. After giving the subdivision rules the authors give the interproximate subdivision algorithm followed by the continuity analysis of the scheme. They prove that the scheme is \(C^2\) continuous only in one special case. Next, the authors introduce the convexity preserving scheme. They modify the interproximate scheme in the way that the new scheme adjusts automatically the tension parameter during the iteration process based on the geometry of the control polygon. After the introduction of the scheme, the authors prove that the proposed scheme is convexity preserving. Then, they make an experimental analysis of the smoothness. For this purpose they numerically check the Hölder regularity of the limit curve. The experiments show that for most of the generated curves the Hölder regularity is between \(2.1\) and \(2.9\). At the end, the authors describe how to modify the interproximate scheme to obtain a better shape behaviour. They let to relax the \(I\) vertices from the interpolation requirement during the iteration process.
scientific article; zbMATH DE number 6138045