In the present paper,we presented the user defined functions and operator overloadings for computing the quotient and remainder obtained by division of two integers in C language environment.These functions and operator overloadings facilitate the users to carry out the operation of division with remainder in C language environment.The result of the running of the program shows the correctness of functions and operator overloadings we defined.

We studied the problem of how to estimate the arc lengths of parametric curves with arbitrary degree in the polynomial form.Compared with the current similar methods,the advantage of our method is that there is no constraint to the degree of the curves.The key problem here is how to calculate all the inflection points of the arc length curve and where to divide the original curve.Using the property of variation diminishing for Bézier curves,we calculated all the inflection points of the arc length curve by Newton iteration method and obtained the approximate arc length curve for the original curve in Bézier form.Numerical experiments show that the results of the arc length curve obtained by our method are satisfactory.

The degree reduction of parameters curves and surfaces is one of the hottest topics in the researches of computer aided geometric design.This paper gave an overview about the methods and the problem of degree reduction for curves and surfaces in recent years.We mainly stated the degree reduction of Bezer curves and surfaces,the others curves and surfaces such as B-splines etc.were also mentioned.

In the present paper,we presented the non-recursive algorithm for finding the greatest common divisor of two integers and its combination coefficients together with the corresponding program in C language environment.Compared with other programs,the program here saves the cost of time and space when the program is running,and so the efficiency is improved.

We presented an algorithm of topological sorting with local collateral property for digraph expressed in adjacency lists in the present paper.Compared with other method,our algorithm has more practicability with little increase in the cost of running time and memory.The related source program and the running results were al-so provided.The results show that our algorithm is feasible.

In order to get an approximation with better effect of parameterization of Bézier curves,we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement,and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result,the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms,the relative deviation will have at least a 20% reduction.

In computer-aided design,transformations among different forms of curves and surfaces are often required to carry out operations of degree-reduction of curves and surfaces and data exchanging between different geometric modeling systems. The errors of these trans-formations would depend on the condition numbers of the corresponding transformation matrices. For this reason,we studied some properties of Jacobi-Bernstein basis transformation matrices related to their condition numbers,and by computing the infinite norms of the transformation matrices and their inverse matrices,we obtained explicit upper bounds to these condition numbers. An example of applications of these condition numbers in CAGD was also provided.

Using de Casteljau algorithm,The allthors derive the sufficient and necessary conditions of a developable surface being a ruled surface generated by curves of n degree and m degree respectively as its two boundaries,point out the number of constraint conditions of a ruled surface,and present a new method of designing ruled surface of(2,3) degree.Experimental results show that the design method proposed is easy to design and easy to control the surface shape.

We provide a method for approximate arc-length parameterization for Bezier curves by using the subdivision techniques and give the corresponding algorithm. By finding the so-called `worst point' of the curve and subdividing the curve at the corresponding parameter value, we got two Bezier curves. Let the two curves have weights that are proportional to their approximate arc-lengths. Then we repeatedly do the same to the newly generated Bezier curves and finally we can get a piecewise Bezier curve. By means of knot inserting technique we convert it into such a curve that has the B-spline form. Now this new curve has a global parameter and in the interval of which each Bezier curve will have a parameter sub-interval of length proportional to its weight. Thus we get a curve with approximate arc-length parameterization

By using reparameterization with a quadratic transformation on the parametric domain,an approximate arc-length parameterization method and the corresponding algorithm are presented for Bézier curves.The key of this algorithm is to choose suitable transformation for preserving the regularity of the given curve.Experiments show that the distribution of the points on the curve has been improved by using the proposed method.

In this paper,we consider the problem of the approximation of nearly arc-length parameterized Bezier curves.When a nearly arc-length parameterization[1] for a Bezier curve has been obtained,the parameterization could only acquire C0-continuity.To increase the parametric continuity,we use its best approximation with constraints at the endpoints with respect to L2-norm to get a Bezier spline curve with C2-continuity.Experiment proves that the effect of this approximation is quite satisfactory.

The algorithm and computer program were given for finding the solutions of linear Diophantine equations in n unknowns by matric method. In con trast to other methods, our method has the following advantages : the algorithm is more simple, easy to be realized and has good commonality.

对双三次Bézier曲面片的光滑拼接条件给出了新的推导.

The paper presents a computer program that solves combinational coefficients of linear expression for the largest common divisor of many integers by matrix method.

运用最优存储思想,给出了Aitken算法的计算机程序,对Aitken算法中所涉及的系数构成的三角矩阵存储方式作了探讨,节约了存储空间.

本文给出了求解二元一次不定方程的计算机程序.

举例说明了P.J.Hilton及U.Stammbacb的<A course in Homological Algebra>中命题7.7中的结论并非普遍成立.