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 delete all the triangles it intersects, define two polygons that fill the so generated hole and have the segment as shared basis, and then re-triangulate each polygon separately. In this paper we prove that the polygons generated evacuating the triangles that intersect a constrained segment 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. In this paper we formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art.

中文翻译：

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使用简化耳切的确定性线性时间约束三角剖分

符合一组不相交输入段的三角剖分算法通常以增量方式进行，首先插入点，然后插入段。插入一个段相当于删除它相交的所有三角形，定义两个多边形来填充如此生成的孔并将该段作为共享基础，然后分别对每个多边形重新三角化。在本文中，我们证明了与约束线段相交的三角形所生成的多边形是这样的：它们的所有凸顶点（除了两个凸顶点）都可以用于以耳切方式形成三角形，而无需检查其他多边形点是否位于每个顶点内耳朵。任何简单多边形至少包含三个凸顶点这一事实保证了有效的耳朵切割的存在，从而确保收敛。这不仅转化为最佳确定性线性时间三角测量算法，而且这种算法实现起来也很简单。在本文中，我们正式证明了我们方法的正确性，并在实际应用中对其进行了验证并将其与现有技术进行了比较。