Two Lepp algorithms for quality Delaunay triangulation are discussed. Firstly a terminal triangles centroid Delaunay algorithm is studied. For each bad quality triangle t, the algorithm uses the longest edge propagating path (Lepp(t)) to find a couple of Delaunay terminal triangles (with largest angles less than or equal to 120 degrees) sharing a common longest (terminal) edge. Then the centroid of the terminal quadrilateral is Delaunay inserted in the mesh. Insertion of the midpoints of some constrained edges are also performed to assure convergence close to the constrained edges. We prove algorithm termination and that a graded, optimal size, 30 degrees triangulation is obtained, for any planar straight line graph (PSLG) geometry with constrained angles greater than or equal to 30 degrees. We also prove that the size of the final triangulation is optimal and that this size is independent of the processing order of the bad triangles in the mesh. Next, by introducing the concept of non-improvable triangles (with constrained angle $<$ 30 degrees), we generalize the algorithm to deal with PSLG geometries with N small constrained angles. Thus given a triangle size parameter $\delta$ for non-improvable triangles, the generalized algorithm constructs a quality triangulation with non constrained angles $\ge$ 30 degrees and at most N non-improvable triangles of size $\delta$ (longest edge $\le$ $\delta$). In practice the algorithms behave as predicted by the theory.

讨论了两种用于质量Delaunay三角剖分的Lepp算法。首先研究了终端三角形质心Delaunay算法。对于每个不良三角形t，该算法使用最长的边缘传播路径（Lepp（t））来找到共享共同的最长（最终）边缘的一对Delaunay最终三角形（最大角度小于或等于120度）。然后将末端四边形的质心插入网格中。还执行一些受约束边缘的中点的插入，以确保收敛于受约束边缘附近。我们证明了算法终止，并且对于约束角大于或等于30度的任何平面直线图（PSLG）几何图形，均获得了渐变的最佳大小30度三角剖分。我们还证明了最终三角剖分的大小是最佳的，并且该大小与网格中不良三角形的处理顺序无关。下一个，$<$30度），我们推广了该算法来处理具有N个小约束角的PSLG几何形状。因此给定三角形大小参数$\delta$ 对于不可改进的三角形，广义算法构造具有不受约束角度的质量三角剖分 $\ge$ 30度，最多N个不可改进的三角形 $\delta$ （最长的边缘 $\le$ $\delta$）。在实践中，算法的行为与理论预测的一样。