A Method for Deducing Interpolating Subdivision Schemes from Approximating Subdivision Schemes and Blending Subdivisions

Abstract

We present a method for deducing interpolating subdivision schemes from known approximating subdivision scheme which generates curves and blending subdivision schemes. We make use of the connection between the approximating subdivision schemes and interpolating subdivision schemes to produce a blending subdivision scheme that integrates interpolating and approximating subdivision schemes in a simple and efficient way. The basic idea is to use the specific linear combination of the displacements of the new vertices which are derived from approximating subdivision scheme to produce associated interpolating displacements for the new vertices therefore generating a corresponding interpolating subdivision scheme. The most importance is that such methods are deduced in premise of keeping the continuity of the original approximating subdivision schemes. In addition, our method has much lower computational complexity compared with the exiting methods and can be easily extended to the case of surface subdivision schemes.

A New Scheme of Interpolation Subdivision Surface by Using the Bezier Curve

“…In Fig.5(a), it illustrates the results of a deducing interpolating subdivision scheme in [14]. While Fig.8 (b) is the results which are obtained by using the subdivision scheme proposed in this paper.…”

“…We utilize the ideas from the NURBS that using the Bezier curve to generate point. Since the 4-point subdivision curve and deducing interpolating subdivision scheme in [14] are both difficult to generate a circle from a square or a triangle, We refresh the mind to propose a scheme to make a curve smooth and circinal, then extend this scheme to the subdivision surface scheme. Meanwhile, it is difficult for the proposed interpolation schemes in [5,10,15] to generate a sphere from a tetrahedron, which has only 4 vertices and 4 faces.…”

“…The final subdivision curves of these two shapes are both precise circles. a) Triangle to Circle b) Square to Circle Fig.4 Generation of a circle a) Deducing interpolating subdivision scheme in [14] b) Subdivision scheme by our methods…”