Circumcenter of Mass and Generalized Euler Line
Circumcenter of Mass and Generalized Euler Line
复制标题
质量圆心和广义欧拉线
DOI：
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发表时间：
2013
影响因子：
0.8
通讯作者：
E. Tsukerman
中科院分区：
文献类型：
--
作者：
S. Tabachnikov;E. Tsukerman
我们定义和研究一个变种的质量中心的多边形，更一般地说，一个单纯的多面体，我们称之为外接圆的质量（CCM）。CCM是多面体三角剖分中单形外心的仿射组合，由它们的体积加权。对于内接多面体，CCM与外心重合。我们的动机来自于完全可积离散动力系统的研究，其中CCM是离散自行车（达布）变换和多边形的recuttings的不变量。我们表明，CCM满足类似的阿基米德引理，一个熟悉的性质的质心。我们定义并研究了一个广义欧拉线相关联的任何单纯多面体，扩展了以前研究的欧拉线相关联的四边形。我们证明了多边形的广义欧拉线由满足自然连续性和齐次性假设以及阿基米德引理的所有中心组成。最后，我们表明，CCM也可以定义在球面和双曲设置。
We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The CCM is an affine combination of the circumcenters of the simplices in a triangulation of a polytope, weighted by their volumes. For an inscribed polytope, CCM coincides with the circumcenter. Our motivation comes from the study of completely integrable discrete dynamical systems, where the CCM is an invariant of the discrete bicycle (Darboux) transformation and of recuttings of polygons. We show that the CCM satisfies an analog of Archimedes’ Lemma, a familiar property of the center of mass. We define and study a generalized Euler line associated to any simplicial polytope, extending the previously studied Euler line associated to the quadrilateral. We show that the generalized Euler line for polygons consists of all centers satisfying natural continuity and homogeneity assumptions and Archimedes’ Lemma. Finally, we show that CCM can also be defined in the spherical and hyperbolic settings.