基于活动标架对Hilbert曲线的研究

Study of Hilbert Curve Based on Moving Frame

现有刻画三维Hilbert曲线的算法大多是从始点到终点递归地计算节点坐标,针对此类算法迭代次数较多的问题,提出一种刻画三维Hilbert曲线的新算法.借助于构造活动标架,得到刚体运动下的不变量,即离散曲率挠率.考虑到活动标架,曲线节点将被重新编码.并建立曲线弯曲点位置编号与其对应的曲率挠率数对的映射,编写相应算法使其对任意编号n,能够输出该编号对应弯曲点的曲率挠率数对且画出弯曲点图象结构.相比于基于Matlab生成Hilbert曲线的算法Hilbert3(n),该算法不局限于曲线的阶数、不依赖相邻阶曲线节点坐标之间的迭代.实验结果表明此算法更加高效.

Most algorithms for describing three-dimensional Hilbert curves calculate node coordinates from the start point to the end point recursively. Directing at the multiple iteration, a new algorithm was brought forth. By means of constructing moving frame, the invariants under rigid body motion are obtained, that is, discrete curvature and torsion. Considering moving frame, the nodes are recoded. Establishing a map between the inflection point location number and the discrete curvature and torsion of the inflection point, based on that, writing the corresponding algorithm to make it for any number n , the pairs of curvature torsion and the image structure corresponding to the inflection points can be output. Compared to the algorithm Hilbert3( n ), the proposed algorithm is not limited to the order of the curve and does not depend on the iteration between the coordinates. Experimental results show that the algorithm is more efficient.