Polyhedral patterns

We study the design and optimization of polyhedral patterns, which are patterns of planar polygonal faces on freeform surfaces. Working with polyhedral patterns is desirable in architectural geometry and industrial design. However, the classical tiling patterns on the plane must take on various shapes in order to faithfully and feasibly approximate curved surfaces. We define and analyze the deformations these tiles must undertake to account for curvature, and discover the symmetries that remain invariant under such deformations. We propose a novel method to regularize polyhedral patterns while maintaining these symmetries into a plethora of aesthetic and feasible patterns.

[1]  Leif Kobbelt,et al.  OpenMesh: A Generic and Efficient Polygon Mesh Data Structure , 2002 .

[2]  Mark Pauly,et al.  Shape‐Up: Shaping Discrete Geometry with Projections , 2012, Comput. Graph. Forum.

[3]  Philippe Bompas,et al.  Planar Panelization with Extreme Repetition , 2014, AAG.

[4]  Johannes Wallner,et al.  Geometric Modeling with Conical Meshes and Developable Surfaces , 2006, ACM Trans. Graph..

[5]  David Bommes,et al.  Rationalization of Triangle‐Based Point‐Folding Structures , 2012, Comput. Graph. Forum.

[6]  Johannes Wallner,et al.  Designing Quad‐dominant Meshes with Planar Faces , 2010, Comput. Graph. Forum.

[7]  Mirela Ben-Chen,et al.  Dupin Meshing: A Parameterization Approach to Planar Hex-Dominant Meshing , 2015 .

[8]  Bailin Deng,et al.  Interactive design exploration for constrained meshes , 2015, Comput. Aided Des..

[9]  Wenping Wang,et al.  Planar Hexagonal Meshing for Architecture , 2015, IEEE Transactions on Visualization and Computer Graphics.

[10]  Pierre Alliez,et al.  Variational shape approximation , 2004, ACM Trans. Graph..

[11]  Peter J Lu,et al.  Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture , 2007, Science.

[12]  Syed Jan Abas,et al.  SYMMETRIES OF ISLAMIC GEOMETRICAL PATTERNS , 1994 .

[13]  Johannes Wallner,et al.  Form-finding with polyhedral meshes made simple , 2014, ACM Trans. Graph..

[14]  Jan Knippers,et al.  Biomimetic Lightweight Timber Plate Shells: Computational Integration of Robotic Fabrication, Architectural Geometry and Structural Design , 2014, AAG.

[15]  Jun Wang,et al.  Freeform Honeycomb Structures , 2014, Comput. Graph. Forum.

[16]  Henrik Zimmer,et al.  Optimization of 3D models for fabrication , 2014 .

[17]  Baining Guo,et al.  General planar quadrilateral mesh design using conjugate direction field , 2011, ACM Trans. Graph..

[18]  A. Bobenko,et al.  Discrete Differential Geometry: Integrable Structure , 2008 .

[19]  Roi Poranne,et al.  Interactive Planarization and Optimization of 3D Meshes , 2013, Comput. Graph. Forum.

[20]  Esan Mandal,et al.  Remeshing Schemes for semi-regular tilings , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[21]  S. Schaefer,et al.  Triangle surfaces with discrete equivalence classes , 2010, ACM Trans. Graph..

[22]  Johannes Wallner,et al.  Packing circles and spheres on surfaces , 2009, ACM Trans. Graph..