Abstract: According to the problem of simplification and reconstruction methods of freedom curve, this paper presents a reverse subdivision scheme based on B-spline of degree three. If the new midpoint and new vertex are determined inappropriately, the simplified curve using the above-mentioned reverse subdivision will not properly retain the characteristic and shape of freedom curve. Thus this paper introduces extreme points and inflexion points as characteristic points of freedom curve, and then presents a kind of reverse subdivision which can retain the characteristic points of freedom curve on the basis of subdivision of B-spline of degree three. By using this reverse subdivision, conformal simplification of freedom curve can be achieved. The freedom curve can be reconstructed by establishing the error vectors while reversing. Finally, this paper presents a practical way of finding the curvature extreme points of discrete point set.

Key words: reverse subdivision, curve subdivision, simplification and reconstruction

摘要： 针对自由曲线的简化和重构问题，在三次B样条细分方法的基础上，提出了相应的逆向细分法，但是如果不能很好地确定曲线中新边点和新顶点，应用该逆向细分方法最终得到的简化控制多边形并不能很好地保持原曲线的一些形状与特征。为了使简化曲线能够更好地保持原有曲线的特征与形状，引入了自由曲线的极值点和拐点作为自由曲线的特征点，进一步提出了保留自由曲线特征点的三次B样条逆向细分法，并将其应用于自由曲线的简化中，可以实现自由曲线的保形简化，并且通过在曲线简化时保留误差向量，实现自由曲线的重构。对如何判别离散点集曲率极值点进行了研究，并给出了相应的简单判别方法。

关键词: 逆向细分, 曲线细分, 简化与重构