Journal of Computational Mathematics, Vol.25, No.4, 2007, 473–484. ESTIMATING ERROR BOUNDS FOR TERNARY SUBDIVISION CURVES/SURFACES *1) Ghulam Mustafa (Department of Mathematics, Islamia University, Bahawalpur, Pakistan Department of Mathematics, University of Science and Technology of China, Hefei 230026, China Email: mustafa rakib@yahoo.com) Jiansong Deng (Department of Mathematics, University of Science and Technology of China, Hefei 230026, China Email: dengjs@ustc.edu.cn ) Abstract We estimate error bounds between ternary subdivision curves/surfaces and their control polygons after k-fold subdivision in terms of the maximal differences of the initial control point sequences and constants that depend on the subdivision mask. The bound is inde- pendent of the process of subdivision and can be evaluated without recursive subdivision. Our technique is independent of parametrization therefore it can be easily and efficiently implemented. This is useful and important for pre-computing the error bounds of subdivi- sion curves/surfaces in advance in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and so on. Mathematics subject classification: 65D17, 65D07, 65D05. Key words: Subdivision curve, Subdivision surface, Subdivision depth, Error bound. 1. Introduction Subdivision is an important method for generating smooth curves and surfaces, see, e.g., [1, 2, 8]. Efficiency of subdivision algorithms, their flexibility and simplicity have found their way into wide applications in Computer Graphics and Computer Aided Geometric Design (CAGD). A widely used, efficient and intuitive way to specify, represent and reason about curved, surfaces, nonlinear geometry for design and modeling is the control polygon paradigm. For many applications, e.g., rendering, intersection testing or design, this raises the question just how well the control polygon approximates the exact curved and surface geometry. Several researchers give several answers to this question. Nairn et al. [7] show that the maximal distance between a B´ ezier segment and its control polygon is bounded in terms of the differences of the control point sequence and a constant that depends only on the degree of the polynomial. Lutterkort and Peters [6] derived a sharp bound on the distance between a spline and its B- spline control polygon. Their bound yields a piecewise linear envelope enclosing the spline and the control polygon. Recently, Karavelas et al. [5] derived sharp bounds for the distance between a planar parametric B´ ezier curve and parameterizations of its control polygon based on the Greville abscissae. In [1], Cheng gave an algorithm to estimate subdivision depths for rational curves and surfaces. The subdivision depth is not estimated for the given curve/surface * Received January 18, 2005; final revised January 24, 2006; accepted June 28, 2006. 1) This work was supported in part by NSF of China (No. 10201030), the TRAPOYT in Higher Education Institute of MOE of China and the Doctoral Program of MOE of China (No. 20010358003).