Research On Accuracy Problems In Continuous And Discrete Geometric Modeling

Geometric modeling approaches can be divided into two categories according to whether they are dependent on function expressions. One is continuous, and the other is discrete. Continuous modeling approach is based on expressions of curves and surfaces. Discrete modeling approach directly gets, according to certain rules, more and more points from the initial data points. A piecewise curve or a piecewise surface, known as the control polygon or control net, is formed when the points are linked topologically. Repeating the process of generation of new control points and new control structures we at last obtain a smooth curve or a smooth surface if the rules are appropriate.Taking into account the geometric features of curves and surfaces and the complexity of the calculations, and many other factors, one in general selects polynomials or piecewise polynomials as an approximation of the original functions in practical applications of continuous modeling approach. In this paper, how to select a suitable approximation and how to analysis the approximation error for the continuous modeling method are the first category of accuracy problems.Since the operations of the discrete modeling method are intuitive, simple and easy interactive control, they are particularly suited for computer processing. On the whole, the limit shapes of curves and surfaces can be judged from their control structures. However, in some practical applications we still want to know the values in the limit curve (surface) with respect to certain parameters for accurate processing or analysis. In addition, control structures after refinement are usually used to replace the curve (surface) which is also a kind of approximation. We also want to know the error between this approximation and the real situation.Hence there are two important problems in the discrete geometric modeling need to be considered. First, how to calculated the value of a point in the curve (surface) corresponding to a parameter value. Second, how to measure the error between the control structures and the limit curve (surface). Here, we call the error estimate problem the second category of accuracy problems.In terms of some mathematical tools, such as functional analysis and operator approximation theory, we discuss the first category of accuracy problems firstly. For the second category of accuracy problems, some traditional techniques, such as eigenanaly- sis and generating function, are used for our discussions, and a new technique, namely, blossoming, which is a very powerful tool in dealing with polynomials is used for some discussions on Bezier curve and B splines. We also give some results on the general subdivision methods for curves. In particular, a B-spline interpolation subdivision algorithm is prosed. A detailed analysis shows that this algorithm is more accuracy than the ordinary subdivision algorithm for B splines. To end up the discussion of the discrete geometric modeling method of one variable, quasi-interpolation techniques which can improve approximation accuracy for more general discrete modeling approaches, with an example, are introduced.Subdivision surface technique is one of the most important discrete modeling techniques for building free-form surfaces. One of the major advantages of such a technique is that it applies to surfaces of arbitrary topology. It has a wide range of applications, such as CAGD, computer graphics and medical image processing. But for many years some unsolved theoretical problems limit their applications in industry. That began to change until the mid 90s of the 20th century. Exact evaluation and error estimate for subdivision surfaces are two representative questions.Exact evaluation for subdivision surface has been solved by Jos Stam, but error estimate problems are still very difficult. We introduce some preliminary results for the famous Catmull-Clark subdivision surface which is a generalization of bi-cubic B-spline surface. In addition, We presented a new exact analytical evaluation formula for Loop subdivision surfaces which are different form the numerical formula of Stam. What's more, by means of the eigenanalysis technique we obtain a precise high power of the subdivision matrix, and further find recurrence formulas for a so-called G-difference, which is a new kind of difference applicable to space quadrilaterals and has an obvious geometric meaning. In terms of these formulas we get the convergence rate for G-differences and establish an error estimate method for Loop subdivision surfaces. Examples and numerical experimental results show that our estimates are optimum and near optimum for the regular case and extraordinary cases, respectively.