中文摘要：

周期B样条基以一种简洁的形式表示闭B样条曲线.周期B样条基转换矩阵为闭B样条曲线及相关曲面的不同表示间的转换提供了一个数学模型.本文给出了周期B样条基转换矩阵的存在性条件,给出并证明了周期B样条基转换矩阵的一个简单的递归表示式.在此基础上,本文进一步给出了周期B样条基转换矩阵的计算公式和高效算法.周期B样条基转换矩阵为闭B样条曲线的节点插入、升阶、节点删除和降阶等基本运算提供了一个统一而简单的解决方法,本文给出了一些应用例子.

英文摘要：

With periodic B-spline bases closed B-spline curves can be represented compactly. The transformation matrices of periodic B-spline bases provide a mathematical model for the conversion between different representations of closed B-pline curves and the relevant surfaces.In this paper,we first present the existence conditions of the transfor- mation matrices of periodic B-spline bases.Then we propose and demonstrate a simple recursive formula for the transformation matrices of periodic B-spline bases.Based on the recursive formula,we further develop an efficient algorithm for the computation of the transformation matrices of periodic B-spline bases.With the transformation matrices of periodic B-spline bases,we can establish a uniform scheme for the essential operations of closed B-spline curves and the relevant surfaces such as knot insertion,knot removal,degree elevation and degree reduction.Several examples for the applications of the transformation matrices of periodic B-spline bases are provided lastly.