确定性线性时间约束三角剖分的简化耳割算法
Deterministic Linear Time Constrained Triangulation using Simplified Earcut
摘要：
符合一组不相交输入线段的三角剖分算法通常以增量方式进行，即先插入点，然后插入线段。插入一个线段相当于：（1）删除它相交的所有三角形;（2）用两个多边形填充这样生成的洞，这两个多边形具有所需的线段作为共享边;（3）分别对每个多边形进行三角剖分。在本文中，我们证明了这些多边形是这样的，他们所有的凸顶点，但两个可以用来形成三角形的耳切方式，而不需要检查其他多边形点是否位于每个耳朵。任何简单多边形至少包含三个凸顶点的事实保证了有效耳的存在，以确保收敛。这不仅转化为最优的确定性线性时间三角剖分算法，而且这种算法实现起来也是微不足道的。我们正式证明了我们的方法的正确性，也验证了它在实际应用中，并与现有技术进行比较。
英文摘要：
Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to: (1) deleting all the triangles it intersects; (2) filling the so generated hole with two polygons that have the wanted segment as shared edge; (3) triangulate each polygon separately. In this paper we prove that these polygons are such that all their convex vertices but two can be used to form triangles in an earcut fashion, without the need to check whether other polygon points are located within each ear. The fact that any simple polygon contains at least three convex vertices guarantees the existence of a valid ear to cut, ensuring convergence. Not only this translates to an optimal deterministic linear time triangulation algorithm, but such algorithm is also trivial to implement. We formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art.
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