# Variational Tangent Plane Intersection for Planar Polygonal Meshing

Several theoretical and practical geometry applications are based on polygon meshes with planar faces. The planar panelization of freeform surfaces is a prominent example from the field of architectural geometry. One approach to obtain a certain kind of such meshes is by intersection of suitably distributed tangent planes. Unfortunately, this simple tangent plane intersection (TPI) idea is limited to the generation of hex-dominant meshes: as vertices are in general defined by three intersecting planes, the resulting meshes are basically duals of triangle meshes.

The explicit computation of intersection points furthermore requires dedicated handling of special cases and degenerate constellations to achieve robustness on freeform surfaces. Another limitation is the small number of degrees of freedom for incorporating design parameters.

Using a variational re-formulation, we equip the concept of TPI with additional degrees of freedom and present a robust, unified approach for creating polygonal structures with planar faces that is readily able to integrate various objectives and constraints needed in different applications scenarios. We exemplarily demonstrate the abilities of our approach on three common problems in geometry processing.