Geometric Processing of Polyhedral Meshes

In the proposed project we study means to create, edit and investigate polyhedral meshes. Such meshes are non- triangular meshes in which all faces are planar. Polyhedral meshes are important in architectural geometric design, as they can be realized with flat plates made of wood or glass, for example. Our recent research of polyhedral-mesh processing focused on investigating linear shape-spaces of mesh deformations, and on the remeshing of triangular meshes into hexagonally-dominant meshes. In this project, we propose to develop three different methods beneficial for the geometry processing of such meshes, and that are explained as follows. First, we define and study discrete polyhedral surface flows for infinitesimal surface editing. Our focus is on mean curvature and Willmore flows. Such flows would allow us to define concepts like discrete polyhedral minimal surfaces and constant mean-curvature surfaces in a new way. This is done by employing continuous deformation in a linear subspace of compatible projective maps. Next, we establish an algorithm for smooth conjugate vector-field design, which is the basis for remeshing surfaces into polyhedral meshes. The algorithm is based on a complex-polynomial view of nonorthogonal vector fields on surfaces, and offers an easy control over smoothness and user constraints, without resorting to involved mixed- integer formulations. Finally, we study vector calculus on polygonal and polyhedral meshes, in order to consistently define discrete- geometric counterparts to shape operators, affine connections and more. These differential concepts, once consistently defined, can be used to study vector fields on non-triangular surfaces, which are important for various applications in geometry processing, such as deformation and parametrization.

This project targeted the development of techniques to produce and edit polyhedral meshes, which are meshes with planar faces. Such meshes are beneficial for architectural design, and are lauded for their mathematical properties. Two directions were taken in the project: creating meshes by designing them from existing triangular meshes, and editing given polyhedral meshes into new ones by deforming them in a fair and desired manner. A completed work on the first direction was published in SGP 2014, winning the best paper award, and a subsequent paper was recently submitted to SIGGRAPH 2015. A work in the second direction was also recently submitted to SIGGRAPH 2015. Other works benefited from the techniques acquired during the project. One such example is the remeshing of hexagonal meshes, and another such example is a work done on general polyhedral patterns. These works were done in collaboration with colleagues from the group in TU Vienna, where the project leader spent the duration of the project, and in collaboration with international colleagues from ETH Zurich, TechnionIIT, Bar-Ilan University, and KAUST.