The Generalization of the 3-Point Interpolatory Subdivision Scheme
and Its Application in Terrain Simulation

Hongchan Zheng, Zhenglin Ye, Hongxing Zhao, and Youming Lei
Department of Applied Mathematics, Northwestern Polytechnical University,
Xi'an, 710072,
P.R.China

Abstract    
Subdivision is now an important subject with many applications in fields including Computer Aided Geometric Design, Computer Graphics, and computer animation due to its efficiency and simplicity. Subdivision scheme define a curve out of an initial control polygon or a surface out of an initial control mesh by subdividing them according to some refining rules recursively. An uniform and stationary 3-point ternary interpolatory scheme with two parameters are proposed and analyzed. Its is simple, but the absence of intuition meaning of the two parameters limits the application of the subdivision scheme in a way. In this paper we first present an improved 3-point ternary interpolatory scheme with one parameter which has distinct geometric meaning. Then its support and the C^k convergence analysis are presented. The relationship between the parameter and the fractal property of the limit curve is described. It can be used to generate smooth curves as well as fractal-like curves. To extend its modelling ability its generalization to some other aspects is also investigated. Finally the method of terrain simulation by using the presented schemes is proposed. A lot of examples show that the implementation of the presented schemes is easy, efficient and fast in terrain simulation due to the local, ternary and simple property of the schemes.