Construction And Improvement Of Interproximate Subdivision Scheme

As an efficient method of modeling curves and surfaces in CAGD, subdivision simplystarts from a given control polygon and recursively refines the polygon according to certainrules. Interproximate subdivision scheme generates curves that interpolate some given ver-tices and approximate the others. We want to introduce four-point interproximate subdivisionscheme, six-point interproximate subdivision scheme and their improved subdivision schemes.The following two sections are the results of this paper.On the one hand, from the study of the four-point interproximate subdivision scheme andin order to achieve the targets of improving the curvature and shape-preserving properties of thecurve, we give five improved subdivision schemes. We add one step of four-point interpolatingsubdivision algorithm to get the improved four-point interproximate subdivision scheme1. Byadding one more step of four-point interpolating subdivision algorithm the improved scheme1add more interpolating points, so the new curve have more characters of the curve generat-ed by the four-point interpolating subdivision scheme and can avoid the fluctuation near theinterpolating points. Due to the different ways of the tangents, we improve the four-point inter-proximate subdivision scheme and the improved four-point interproximate subdivision scheme1to get three improved four-point interproximate subdivision schemes which can get curveswith better shape preserving. Based on the artifact-free property of the four-point interpolatingsubdivision scheme, we get the improved four-point interproximate subdivision scheme5whichcan generate the curve with better curvature and vision. Those five schemes are the interproxi-mate subdivision schemes and their eigenvalues of the refinement matrix satisfy the necessarycondition of C2continuity.On the other hand, we combine the six-point interpolating subdivision scheme and thecubic B-spline refinement algorithm to the six-point interproximate subdivision scheme. Com-pared with the four-point interproximate subdivision scheme, it ensures that eigenvalues of therefinement matrix satisfy the necessary condition of C2continuity with smaller range of theparameter. And it can generate curves with better quality and curvature than the four-pointinterproximate subdivision scheme when their parameters are equal. Then we improve thesix-point interproximate subdivision scheme by adding one more step six-point interpolatingsubdivision iteration. Through improving we get more interpolating points so the curves get-ting by it contain more characters of six-point interpolating subdivision curves and have bettervisual looking.