Eurographics Symposium on Geometry Processing 2012 Eitan Grinspun and Niloy Mitra (Guest Editors) Volume 31 (2012), Number 5 Modeling Polyhedral Meshes with Affine Maps Amir Vaxman TU Vienna avaxman@geometrie.tuwien.ac.at Abstract We offer a framework for editing and modeling of planar meshes, focusing on planar quad, and hexagonal- dominant meshes, which are held in high demand in the field of architectural design. Our framework manipulates these meshes by affine maps that are assigned per-face, and which naturally ensure the planarity of these faces throughout the process, resulting in a linear subspace of compatible planar deformations for any given mesh. Our modeling metaphors include classical handle-based editing, mesh interpolation, and shape-space exploration, all of which allow for an intuitive way to produce new polyhedral and near-polyhedral meshes by editing. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Picture/Computational Geometry and Object Modeling—Shape modeling, Shape editing Shape space, PQ/PHex meshes, design explo- ration 1. Introduction Meshes with planar faces (classically and hereby denoted as polyhedral meshes) have recently come into prominence, for their benefits in the field of freeform architectural design, where they can be manufactured with relative ease. Designers and modelers thus require tools in which they can manipulate these planar meshes freely and intuitively, while maintaining the planarity constraints. There are several mesh modeling metaphors that designers would usually utilize. A common approach is handle-driven editing, in which vertex displacements, and a given Region-of-interest (ROI) are supplied, and the result mesh adheres to these constraints, while minimizing a set of fairness energies, such as the rigidity, or similarity, of edited faces with relation to the original mesh, fairness of curves on the surface, et cetera. Positional constraints, and sometimes scaling or rotational constraints can usually be incorporated into such systems with ease. A deformation tool is most often equipped with an interpolation tool, that allows the designer to navigate the range of middle shapes between two or more boundary shapes. Recently, designers have expressed interest in the ability to explore the shapes within the continuous range of a given initial mesh, and within certain compatibility conditions and fairness measures, without explicitly determining positional constraints. (a) Original (b) As-similar-as-Possible (c) Shape Space Tangent Exploration (d) Our Deformation Figure 1: Comparison between several methods. On the up- per right is a generalization of as-similar-as-possible defor- mation (in the spirit of [SA07]), which produces a smooth re- sult, but does not preserve planarity at all. On the lower left, Planar shape space exploration by [YYPM11], which uses tangent space vectors for handle-driven exploration, and, thus, preserves planarity only up to first order. Our method, on the lower right, clearly preserves planarity, while still producing a smooth and intuitive result. c  2012 The Author(s) Computer Graphics Forum c  2012 The Eurographics Association and Blackwell Publish- ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. DOI: 10.1111/j.1467-8659.2012.03170.x