发表机构
TU Berlin（柏林工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究定义四边形的离散调和中心，证明其与顶点数值无关、具有莫比乌斯协变性，可推广至多面体，且切线/圆内接四边形的中心有简单闭式表达。
AI 中文摘要
通过将简单四边形的所有顶点连接到一个额外的点对其进行三角剖分，若顶点带有数值，则可给分段线性函数赋予狄利克雷能量。研究表明，作为插入点位置函数的最小狄利克雷能量是凸的，且最小值的位置与顶点处的数值无关——四边形具有离散调和中心，其特征为插入边两侧的电流达到平衡。进一步发现，对合莫比乌斯变换（用于交换四边形对角顶点）的不动点是该能量的临界点，因此离散调和中心具有莫比乌斯协变性。对于切线四边形和圆内接四边形，该中心存在与圆心相关的简单闭式表达。该中心及其与数值无关的特性可推广到d维空间中具有d+2个顶点的多面体，但共形特征仅适用于平面上的四个点。
英文摘要
Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.
