Angles of Arc-Polygons and Lombardi Drawings of Cacti
Angles of Arc-Polygons and Lombardi Drawings of Cacti
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DOI：
10.1016/j.comgeo.2023.101982
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发表时间：
2021-07
期刊：
影响因子：
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通讯作者：
D. Eppstein;Daniel Frishberg;Martha C. Osegueda
中科院分区：
文献类型：
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作者：
D. Eppstein;Daniel Frishberg;Martha C. Osegueda
我们描述了具有圆弧边的非自相交三角形中可能的内角三元组，并且证明了只要所有角度都≤π，就可以通过具有圆弧边的非自相交多边形来实现给定的循环角度序列。作为这些结果的结果，我们证明每个仙人掌都有一个平面 Lombardi 绘图（边缘描绘为圆弧的绘图，在每个顶点以相等的角度相交），以实现其自然嵌入，其中仙人掌的每个周期都是绘图的一个面。然而，存在没有平面 Lombardi 绘图的仙人掌平面嵌入。
We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are ≤π. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.