The Discrete Harmonic Center of a Quadrilateral

2026-09-01

AI Summary

本文探讨了通过在四边形内插入一个点并最小化Dirichlet能量来确定该点的最佳位置，定义了四边形的离散谐波中心，并展示了其与Möbius变换的关系。

Abstract

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.

Problem

• Discrete Harmonic Center
• Quadrilateral
• Dirichlet Energy
• Möbius Covariance

Innovation

• Discrete Harmonic Center
• Dirichlet Energy
• Möbius Covariance
• Quadrilateral Triangulation