### Abstract

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the "roughness") is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa's result.

Original language | English (US) |
---|---|

Pages (from-to) | 651-664 |

Number of pages | 14 |

Journal | Discrete and Computational Geometry |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

- Dirichlet energy
- Laplacian
- Rippa
- Triangulations
- Weighted Delaunay triangulation

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

**A monotonicity property for weighted delaunay triangulations.** / Glickenstein, David A.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 38, no. 4, pp. 651-664. https://doi.org/10.1007/s00454-007-9009-y

}

TY - JOUR

T1 - A monotonicity property for weighted delaunay triangulations

AU - Glickenstein, David A

PY - 2007

Y1 - 2007

N2 - Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the "roughness") is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa's result.

AB - Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the "roughness") is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa's result.

KW - Dirichlet energy

KW - Laplacian

KW - Rippa

KW - Triangulations

KW - Weighted Delaunay triangulation

UR - http://www.scopus.com/inward/record.url?scp=37249009184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37249009184&partnerID=8YFLogxK

U2 - 10.1007/s00454-007-9009-y

DO - 10.1007/s00454-007-9009-y

M3 - Article

AN - SCOPUS:37249009184

VL - 38

SP - 651

EP - 664

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -