基于相伴方法的平面域曲线边界中轴线计算

Computation of medial axis of planar domains with curved boundaries based on Cesaro′s approach

从生成中轴线的几何学原理出发,应用微分几何学中的活动标架和相伴曲线方法,将两条边界曲线与其中轴线建立起法向等距映射关系,形成两对相伴曲线,进而建立起平面域曲线边界中轴变换的几何学模型.研究了边界曲线与其中轴线的位置对应关系、尺度变换关系及微分不变量之间的内在联系,在此基础上,利用已知的边界曲线和初始值,提出一种计算中轴线的跟踪算法.算法直接使用准确的自由曲线描述边界,克服了多边形逼近算法的拓扑结果奇异性;同时不需要迭代,计算效率高,可以实现中轴线的精确快速计算.

From the geometrical theory of the medial axis generation,the normal equidistant mapping relationship between the two boundaries and the medial axis is proposed based on the moving Frenet frames and Cesaro＇s approach of the differential geometry. The two pairs of adjoint curves were formed and the geometrical model of the medial axis transform of the planar domains with curved boundaries was established. The relations of position mapping, scale transform and the differential invariants between the curved boundaries and the medial axis were investigated. Based on this model, a tracing algorithm for the computation of the medial axis was generated by known boundary curve and initials. This algorithm overcomes the topological singularity of the polygon approximation algorithms by using exact curved boundaries, and doesntt need iteration. So, it can be used for the computation of the medial axis effectively and accurately.