The Dirichlet energy of a smooth function measures how variable the function is. Due to its deep connection to the Laplace–Beltrami operator, Dirichlet energy plays an important role in digital geometry processing. Given a 2-manifold triangle mesh $M$ with vertex set $V$, the generalized Rippa’s theorem shows that the Dirichlet energy among all possible triangulations of $V$ arrives at its minimum on the intrinsic Delaunay triangulation (IDT) of $V$. Recently, Delaunay meshes (DM) – a special type of triangle mesh whose IDT is the mesh itself – were proposed, which can be constructed by splitting mesh edges and refining the triangulation to ensure the Delaunay condition. This paper focuses on Dirichlet energy for functions defined on DMs. Given an arbitrary function $f$ defined on the original mesh vertices $V$, we present a scheme to assign function values to the DM vertices $V_{new}\supset V$ by interpolating $f$. We prove that the Dirichlet energy on DM is no more than that on the IDT. Furthermore, among all possible functions defined on $V_{new}$ by interpolating $f$, our scheme attains the global minimum of Dirichlet energy on a given DM.

光滑函数的Dirichlet能量测量函数的可变性。由于其与Laplace–Beltrami运算符的紧密联系，Dirichlet能量在数字几何处理中起着重要的作用。给定2流形三角形网格$\mathrm{中号}$ 带有顶点集 $V$，广义Rippa定理表明，在所有可能的三角剖分中Dirichlet能量 $V$ 到达内在的Delaunay三角剖分（IDT）的最小值 $V$。最近，提出了Delaunay网格（DM）–一种特殊的三角形网格，其IDT为网格本身–可以通过分割网格边缘并细化三角剖分以确保Delaunay条件来构造。本文着重讨论DM定义的函数的Dirichlet能量。给定任意函数$F$ 在原始网格顶点上定义 $V$，我们提出了一种将函数值分配给DM顶点的方案 $V_{ñËw}\supset V$ 通过插值 $F$。我们证明DM上的Dirichlet能量不超过IDT上的Dirichlet能量。此外，在$V_{ñËw}$ 通过插值 $F$，我们的方案在给定的DM上实现了Dirichlet能量的全局最小值。