Curve Subdivision Modelling And Its Applications

The subdivision methods have been one of the hot topics in the field of computer aided geometric design because the subdivision methods can produce well-performed curves. This dissertation introduces three types of effective subdivision schemes, including the approximation subdivision schemes, the subdivision schemes blending the interpolation and approximation, and the interproximate schemes, and studies the convexity, smoothness and other properties of the subdivision algorithms.First of all, we present an approximating subdivision, which is obtained by introducing an offset variable to the interpolating four-point ternary subdivision schemes proposed by Hassan. The new scheme unifies the ternary approximating subdivision and interpolation subdivision. The uniform convergence and Ck smoothness of the limit curve produced by our subdivision scheme are analyzed by means of the Laurent polynomials. Secondly, a stationary quaternary four-point approximating subdivision scheme is presented using cubic B-spline basic functions, The Laurent polynomial is used to discuss the uniform convergence and continuity of the subdivision scheme. At last, we present a new subdivision called four-point interpolatory corner-cutting subdivision, which can be used to generate the limit curves that interpolate some specific vertices and approximate the other vertices. The four-point interpolatory corner-cutting subdivision combines the four-point interpolatory subdivision with the corner-cutting subdivision, where only the vertices specified to be interpolated are fixed and the other vertices are updated at each subdivision step, and is different from either the four-point interpolatory subdivision or corner-cutting subdivision. The limit curves generated by four-point interpolatory corner-cutting subdivision are both monotonicity preserving and convexity preserving.