Freeform Honeycomb Structures
Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion‐free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in de...
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| Published in | Computer graphics forum Vol. 33; no. 5; pp. 185 - 194 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Blackwell Publishing Ltd
01.08.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0167-7055 1467-8659 |
| DOI | 10.1111/cgf.12444 |
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| Abstract | Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion‐free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail “honeycomb structures”, which are defined as torsion‐free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss‐Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e.g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e.g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry. |
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| AbstractList | Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion‐free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail “honeycomb structures”, which are defined as torsion‐free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss‐Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e.g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e.g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry. Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion-free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail "honeycomb structures", which are defined as torsion-free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss-Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e.g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e.g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry. [PUBLICATION ABSTRACT] |
| Author | Jiang, Caigui Wang, Jun Pottmann, Helmut Wallner, Johannes |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References | Bommes D., Zimmer H., Kobbelt L.: Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (2009), #77, 1-10. Proc. SIGGRAPH. doi:10.1145/1531326.1531383. Wang J., Jiang C., Bompas P., Wallner J., Pottmann H.: Discrete line congruences for shading and lighting. Computer Graphics Forum 32 (2013), 53-62. Proc. Symp. Geom. Processing. doi:10.1111/cgf.12172. Zadravec M., Schiftner A., Wallner J.: Designing quad-dominant meshes with planar faces. Computer Graphics Forum 29 (2010), 1671-1679. Proc. Symp. Geom. Processing. doi:10.1111/j.1467-8659.2010.01776.x. Bo P., Pottmann H., Kilian M., Wang W., Wallner J.: Circular arc structures. ACM Trans. Graph. 30 (2011), #101, 1-11. Proc. SIGGRAPH. doi:10.1145/1964921.1964996. Zimmer H., Campen M., Bommes D., Kobbelt L.: Rationalization of triangle-based point-folding structures. Computer Graphics Forum 31 (2012), 611-620. Proc. Eurographics. doi:10.1111/j.1467-8659.2012.03040.x. Liu L., Xu Y., Zhang L., Gotsman C., Gortler S.J.: A Local/Global Approach to Mesh Parameterization. Computer Graphics Forum 27, 5 (2008), 1495-1504. doi:10.1111/j.1467-8659.2008.01290.x. Tang C., Sun X., Gomes A., Wallner J., Pottmann H.: Form-finding with polyhedral meshes made simple. ACM Trans. Graphics 33, 4 (2014). Proc. SIGGRAPPH. doi:10.1145/2601097.2601213. Fu C.-W., Lai C.-F., He Y., Cohen-Or D.: K-set tilable surfaces. ACM Trans. Graph. 29 (2010), #44, 1-6. Proc. SIGGRAPH. doi:10.1145/1833349.1778781. Nieser M., Palacios J., Polthier K., Zhang E.: Hexagonal global parameterization of arbitrary surfaces. IEEE Trans. Vis. Comp. Graphics 18, 6 (2012), 865-878. doi:10.1109/TVCG.2011.118. Lu L., Sharf A., Zhao H., Wei Y., Fan Q., Chen X., Savoye Y., Tu C., Cohen-Or D., Chen B.: Build-to-last: Strength to weight 3D printed objects. ACM Trans. Graph. 33, 4 (2014). Proc. SIGGRAPH. doi:10.1145/2601097.2601168. Schiftner A., Höbinger M., Wallner J., Pottmann H.: Packing circles and spheres on surfaces. ACM Trans. Graph. 28 (2009), #139, 1-8. Proc. SIGGRAPH Asia. doi:10.1145/1618452.1618485. Pottmann H., Liu Y., Wallner J., Bobenko A., Wang W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph. 26 (2007), #65, 1-11. Proc. SIGGRAPH. doi:10.1145/1275808.1276458. Singh M., Schaefer S.: Triangle surfaces with discrete equivalence classes. ACM Trans. Graph. 29 (2010), #46, 1-7. Proc. SIGGRAPH. doi:10.1145/1778765.1778783. Song P., Fu C.-W., Goswami P., Zheng J., Mitra N., Cohen-Or D.: Reciprocal frame structures made easy. ACM Trans. Graph. 32 (2013), #94, 1-10. Proc. SIGGRAPH. doi:10.1145/2461912.2461915. 2011; 30 2012; 18 2014 2013 2013; 32 2010; 29 2014; 33 2012; 31 2008; 27 2007; 26 2009; 28 e_1_2_8_16_2 e_1_2_8_17_2 e_1_2_8_12_2 e_1_2_8_13_2 e_1_2_8_14_2 e_1_2_8_15_2 e_1_2_8_9_2 e_1_2_8_2_2 e_1_2_8_4_2 e_1_2_8_3_2 e_1_2_8_6_2 e_1_2_8_5_2 e_1_2_8_8_2 e_1_2_8_7_2 e_1_2_8_10_2 e_1_2_8_11_2 |
| References_xml | – reference: Wang J., Jiang C., Bompas P., Wallner J., Pottmann H.: Discrete line congruences for shading and lighting. Computer Graphics Forum 32 (2013), 53-62. Proc. Symp. Geom. Processing. doi:10.1111/cgf.12172. – reference: Lu L., Sharf A., Zhao H., Wei Y., Fan Q., Chen X., Savoye Y., Tu C., Cohen-Or D., Chen B.: Build-to-last: Strength to weight 3D printed objects. ACM Trans. Graph. 33, 4 (2014). Proc. SIGGRAPH. doi:10.1145/2601097.2601168. – reference: Schiftner A., Höbinger M., Wallner J., Pottmann H.: Packing circles and spheres on surfaces. ACM Trans. Graph. 28 (2009), #139, 1-8. Proc. SIGGRAPH Asia. doi:10.1145/1618452.1618485. – reference: Song P., Fu C.-W., Goswami P., Zheng J., Mitra N., Cohen-Or D.: Reciprocal frame structures made easy. ACM Trans. Graph. 32 (2013), #94, 1-10. Proc. SIGGRAPH. doi:10.1145/2461912.2461915. – reference: Nieser M., Palacios J., Polthier K., Zhang E.: Hexagonal global parameterization of arbitrary surfaces. IEEE Trans. Vis. Comp. Graphics 18, 6 (2012), 865-878. doi:10.1109/TVCG.2011.118. – reference: Tang C., Sun X., Gomes A., Wallner J., Pottmann H.: Form-finding with polyhedral meshes made simple. ACM Trans. Graphics 33, 4 (2014). Proc. SIGGRAPPH. doi:10.1145/2601097.2601213. – reference: Liu L., Xu Y., Zhang L., Gotsman C., Gortler S.J.: A Local/Global Approach to Mesh Parameterization. Computer Graphics Forum 27, 5 (2008), 1495-1504. doi:10.1111/j.1467-8659.2008.01290.x. – reference: Fu C.-W., Lai C.-F., He Y., Cohen-Or D.: K-set tilable surfaces. ACM Trans. Graph. 29 (2010), #44, 1-6. Proc. SIGGRAPH. doi:10.1145/1833349.1778781. – reference: Bo P., Pottmann H., Kilian M., Wang W., Wallner J.: Circular arc structures. ACM Trans. Graph. 30 (2011), #101, 1-11. Proc. SIGGRAPH. doi:10.1145/1964921.1964996. – reference: Singh M., Schaefer S.: Triangle surfaces with discrete equivalence classes. ACM Trans. Graph. 29 (2010), #46, 1-7. Proc. SIGGRAPH. doi:10.1145/1778765.1778783. – reference: Zimmer H., Campen M., Bommes D., Kobbelt L.: Rationalization of triangle-based point-folding structures. Computer Graphics Forum 31 (2012), 611-620. Proc. Eurographics. doi:10.1111/j.1467-8659.2012.03040.x. – reference: Pottmann H., Liu Y., Wallner J., Bobenko A., Wang W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph. 26 (2007), #65, 1-11. Proc. SIGGRAPH. doi:10.1145/1275808.1276458. – reference: Zadravec M., Schiftner A., Wallner J.: Designing quad-dominant meshes with planar faces. Computer Graphics Forum 29 (2010), 1671-1679. Proc. Symp. Geom. Processing. doi:10.1111/j.1467-8659.2010.01776.x. – reference: Bommes D., Zimmer H., Kobbelt L.: Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (2009), #77, 1-10. Proc. SIGGRAPH. doi:10.1145/1531326.1531383. – volume: 33 start-page: 4 year: 2014 article-title: Build‐to‐last: Strength to weight 3D printed objects publication-title: ACM Trans. Graph – volume: 28 start-page: 1 issue: 3 year: 2009 end-page: 10 article-title: Mixed‐integer quadrangulation publication-title: ACM Trans. Graph – volume: 29 start-page: 1671 year: 2010 end-page: 1679 article-title: Designing quad‐dominant meshes with planar faces publication-title: Computer Graphics Forum – volume: 26 start-page: 1 issue: #65 year: 2007 end-page: 11 article-title: Geometry of multi‐layer freeform structures for architecture publication-title: ACM Trans. Graph – start-page: 494 year: 2013 – volume: 33 start-page: 4 year: 2014 article-title: Form‐finding with polyhedral meshes made simple publication-title: ACM Trans. Graphics – volume: 32 start-page: 53 year: 2013 end-page: 62 article-title: Discrete line congruences for shading and lighting publication-title: Computer Graphics Forum – volume: 29 start-page: 1 issue: #46 year: 2010 end-page: 7 article-title: Triangle surfaces with discrete equivalence classes publication-title: ACM Trans. Graph – volume: 31 start-page: 611 year: 2012 end-page: 620 article-title: Rationalization of triangle‐based point‐folding structures publication-title: Computer Graphics Forum – volume: 18 start-page: 865 issue: 6 year: 2012 end-page: 878 article-title: Hexagonal global parameterization of arbitrary surfaces publication-title: IEEE Trans. Vis. Comp. Graphics – volume: 29 start-page: 1 issue: #44 year: 2010 end-page: 6 article-title: K‐set tilable surfaces publication-title: ACM Trans. Graph – volume: 28 start-page: 1 issue: #139 year: 2009 end-page: 8 article-title: Packing circles and spheres on surfaces publication-title: ACM Trans. Graph – year: 2014 – volume: 27 start-page: 1495 issue: 5 year: 2008 end-page: 1504 article-title: A Local/Global Approach to Mesh Parameterization publication-title: Computer Graphics Forum – volume: 30 start-page: 1 issue: #101 year: 2011 end-page: 11 article-title: Circular arc structures publication-title: ACM Trans. Graph – volume: 32 start-page: 1 issue: #94 year: 2013 end-page: 10 article-title: Reciprocal frame structures made easy publication-title: ACM Trans. Graph – ident: e_1_2_8_7_2 doi: 10.1109/TVCG.2011.118 – ident: e_1_2_8_6_2 doi: 10.1111/j.1467-8659.2008.01290.x – ident: e_1_2_8_11_2 doi: 10.1145/2461912.2461915 – ident: e_1_2_8_16_2 doi: 10.1111/j.1467–8659.2012.03040.x – ident: e_1_2_8_15_2 doi: 10.1111/cgf.12172 – ident: e_1_2_8_9_2 doi: 10.1145/1275808.1276458 – ident: e_1_2_8_13_2 doi: 10.1145/1778765.1778783 – ident: e_1_2_8_17_2 doi: 10.1111/j.1467–8659.2010.01776.x – ident: e_1_2_8_12_2 doi: 10.1145/1618452.1618485 – ident: e_1_2_8_3_2 doi: 10.1145/1531326.1531383 – ident: e_1_2_8_14_2 doi: 10.1145/2601097.2601213 – ident: e_1_2_8_5_2 doi: 10.1145/2601097.2601168 – ident: e_1_2_8_4_2 doi: 10.1145/1833349.1778781 – ident: e_1_2_8_8_2 doi: 10.1016/j.cad.2014.02.009 – ident: e_1_2_8_10_2 – ident: e_1_2_8_2_2 doi: 10.1145/1964921.1964996 |
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| SubjectTerms | Analysis Categories and Subject Descriptors (according to ACM CCS) Computer graphics Finite element method Geometry Honeycomb Honeycomb construction Honeycomb structures I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms Industrial engineering Shading Smoothness Studies Topological manifolds Walls |
| Title | Freeform Honeycomb Structures |
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