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Abstract

Islamic geometric patterns known as “girihs” are typically seen as 2D designs. However, in traditional Iranian architecture, girihs have been used to decorate curved and 3D surfaces. While there have been many studies on 2D girihs and methods for creating them, there has been less focus on 3D girihs. Additionally, there isn’t a comprehensive digital technique for constructing 3D girihs. This paper aims to fill this gap by proposing an automated approach for constructing girihs on curved surfaces to create 3D girihs. We use the UV coordinate relative to the NURBS (Non-Uniform Rational B-Spline) surfaces to identify the corresponding generative points of 2D girihs on 3D surfaces, then connect the points to construct 3D girihs. With this method, any geometric girih can be mapped onto a 3D surface, and under the right conditions, any 2D designs and motifs can also be mapped. We implement the proposed method in Grasshopper for Rhino 8 for Win, controlling the density and mapping quality of 3D girihs through parameters N_MU and DL. By integrating this algorithm with a 2D girih generation algorithm, we provide the possibility of real-time customization and interactive construction of 3D girihs.

Keywords

Islamic geometric patterns 3D girihs NURBS surfaces UV coordinate

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References

Pottmann

Eigensatz

Vaxman

, et al. Architectural geometry. Comput Graph 2014; 47: 145–164. DOI: 10.1016/j.cag.2014.11.002.

Senagala

. Rethinking smart architecture: some strategic design frameworks. Int J Architect Comput 2006; 4: 33–46.

Brodey

. The design of intelligent environments: soft architecture. Landscape 1967; 17: 8–12.

Negroponte

. Toward a theory of architecture machines. J Architect Educ 1969; 23: 9–12.

Abas

Salman

. Symmetries of Islamic geometrical patterns. Singapore: World Scientific, 1994.

Wichmann

. Symmetry in Islamic geometric art. Symmetry Cult Sci 2008; 19(2–3): 95–112.

Grünbaum

Shephard

Grunbaum

. Interlace patterns in Islamic and moorish art. Leonardo 1992; 25(3): 331–339.

Djibril

Thami

ROH

. Islamic geometrical patterns indexing and classification using discrete symmetry groups. J Comput Cult Herit 2008; 1(2): 1–14. DOI: 10.1145/1434763.1434767.

Albert

Gómis

Blasco

, et al. A new method to analyse mosaics based on symmetry group theory applied to Islamic geometric patterns. Comput Vis Image Underst 2015; 130: 54–70. DOI: 10.1016/j.cviu.2014.09.002.

10.

El-Said

Parman

. Geometrical concepts in islamic art. London, UK: World of Islam Festival. Publ. Co, 1976.

11.

Bonner

. Islamic geometric patterns; their historical development and traditional methods of construction. 1st ed. New York, NY: Springer, 2017.

12.

Hankin

. The drawing of geometric patterns in saracenic art. In: Volume 15 of memoirs of the archaeological society of India. New Delhi, India: Government of India, 1925.

13.

Hankin

. Examples of methods of drawing geometrical arabesque patterns. Aust Math Soc Gaz 1925: 371–373.

14.

Hankin

. Some difficult saracenic designs. Math Gaz 1934; 18: 165–168.

15.

Hankin

. Some difficult saracenic designs. III: a pattern containing fifteen-rayed stars. Math Gaz 1936; 20: 318–319.

16.

Kaplan

. Islamic star patterns from polygons in contact. In: 2005 Proceedings of graphics interface. GI’05, Waterloo, ON, 9–11 May 2005. Canadian HumanComputer Communications Society, School of Computer Science, University of Waterloo.

17.

Kaplan

. Computer generated Islamic star patterns. In: Proceedings of graphics interface GI’05, Waterloo, ON, 9–11 May 2005. Canadian HumanComputer Communications Society, School of Computer Science, University of Waterloo.

18.

Kaplan

Salesin

. Islamic star patterns in absolute geometry. ACM Trans Graph 2004; 23(2): 97–119. DOI: 10.1145/990002.990003.

19.

Azizi Naserabad

Ghanbaran

. Presenting a method to generate existing and novel girihs (Islamic geometric patterns) for all systems and families based on 7-fold patterns. Comput Aided Des 2023; 154: 103418. DOI: 10.1016/j.cad.2022.103418.

20.

Azizi Naserabad

Ghanbaran

. Generating novel 6-point girihs (6-point Islamic geometric patterns) based on existing ones. SN Comput Sci 2024; 5: 611. DOI: 10.1007/s42979-024-02964-5.

21.

Azizi Naserabad

Ghanbaran

. Computational approach in presentation a parametric method to construct hybrid girihs (hybrid Islamic geometric patterns). Int J Architect Comput. 2024; 14780771241279347. doi:10.1177/14780771241279347.

22.

Khamjane

Benslimane

. A computerized method for generating Islamic star patterns. Comput Aided Des 2018; 97: 15–26.

23.

Khamjane

Benslimane

Ouazene

. Method of construction of decagonal self-similar patterns. Nexus Netw J 2020; 22: 507–520.

24.

Bonner

. The historical significance of the geometric designs in the northeast dome chamber of the Friday Mosque at Isfahan. Nexus Netw J 2016; 18: 55–103.

25.

Bodner

. A nine-and twelve-pointed star polygon design of the Tashkent scrolls. In: Bridges 2011: mathematics, music, art, architecture, culture, Pécs, Hungary, 27–28 July 2011, pp. 147–154.

26.

Bodner

. From sultaniyeh to tashkent scrolls: euclidean constructions of two nine and twelve-pointed interlocking star polygon designs. Nexus Netw J 2012; 14: 307–332. DOI: 10.1007/s00004-012-0111-y.

27.

Bodner

. Bourgoin’s 14-pointed star polygon designs. In: Bridges 2010: mathematics, music, art, architecture, culture, Pécs, Hungary, 24–29 July 2010, 135–142.

28.

Bodner

. Hankin’s ‘polygons in contact’ grid method for recreating a decagonal star polygon design. In: Sarhangi

Séquin

(eds) Bridges Leeuwarden: mathematics, music, art, architecture, culture. London, UK: Tarquin Publications, 2008, pp. 21–28. Available online at: https://archive.bridgesmathart.org2008/bridges2008-21.html.

29.

Bodner

. Construction and classifying designs of al-andalus. In: JAMJM-ARS

Nathaniel

Séquin

(eds) Meeting Alhambra, ISAMA-BRIDGES conference proceedings. Granada, Spain: University of Granada, 2003, pp. 61–68. Available online at: https://archive.bridgesmathart.org2003/bridges2003-61.html.

30.

Lee

. Islamic star patterns. Muqarnas 1987; 4: 182–197.

31.

Heckbert

. Survey of texture mapping. IEEE Comput Graph Appl 1986; 6: 56–67.

32.

Hughes

Dam

McGuire

, et al. Computer graphics - principles and practice. 3rd ed. Boston, MA: Addison-Wesley, 2014.

33.

Shreiner

Woo

Neider

, et al. OpenGL(R) programming guide: the official guide to learning OpenGL(R). 8th ed. Boston, MA: Addison-Wesley, 2005.

34.

Sun

Bao

. Interactive texture mapping for polygonal models. Comput Geom 2000; 15: 41–49.

35.

Maillot

Yahia

Verroust-Blondet

. Interactive texture mapping. In: Proceedings of the 20th annual conference on computer graphics and interactive techniques, Anaheim CA, 2–6 August 1993.

36.

How do you effectively create a UV map for a character model?

https://www.linkedin.com/advice/0/how-do-you-effectively-create-uv-map-character-model#:∼:text=Choosea_suitable_software_2_Prepare_your_model,4_Pack_your_UVs_into_a_UV_tile.

37.

Calvello

. What is UV mapping? How it makes 3D models come to life. https://www.g2.com/articles/uv-mapping (2022).

38.

Robison

. UV mapping explained. https://myndworkshop.com/resources/uv-mapping-explained.

39.

UV mapping. https://www.rhino3d.com/features/rendering/uv-mapping.

40.

Iwai

. Projection mapping technologies: a review of current trends and future directions. Proc Jpn Acad Ser B Phys Biol Sci 2024; 100: 234–251.

41.

Projection mapping: what it is and the simplest way to do it!

https://www.heavym.net/what-projection-mapping-is-and-how-to-do-it/.

42.

Bernik

Kotlar

. Ptex system for advanced texture mapping. Acta Graph 2018; 29: 15–24. DOI: 10.25027/agj2017.28.v29i1.135.

43.

Burley

Lacewell

. Ptex: per-face texture mapping for production rendering. Comput Graph Forum 2008; 27: 1155–1164. DOI: 10.1111/j.1467-8659.2008.01253.x.

44.

Weiss

Bayer

Westermann

. Triplanar displacement mapping for terrain rendering. Eurographics. 2020.

45.

Kasraei

Nourian

Mahdavinejad

. Girih for domes: analysis of three Iranian domes. Nexus Netw J 2016; 18: 311–321.

46.

Ebrahimi

Shoubi

. The projection strategies of gireh on the Iranian historical domes. Mathe Interdiscip Res 2020; 5(3): 239–257. DOI: 10.22052/mir.2020.212903.1187.

47.

Cromwell

. Creating star patterns on the sphere. 2023. DOI: 10.13140/RG.2.2.12421.32489.

48.

Bonner

. Doing the jitterbug with Islamic geometric patterns. J Math Arts 2018; 12(2–3): 128–143. DOI: 10.1080/17513472.2018.1466431.

49.

Dunham

. Hyperbolic Islamic patterns–a beginning. London, UK: Bridges, 2001, 247–254.

50.

Sayed

Ugail

Palmer

, et al. Auto-parameterized shape grammar for constructing islamic geometric motif-based structures. Trans Comput Sci 2016; 28: 146–162.

51.

Safaeianpour

Valibeig

. A study on the construction technology of the Seljuk minarets in Isfahan with focus on their geometric brick pattern. Curved Layer Struct 2021; 9: 13–24.

52.

Kniat

. The quick measure of a nurbs surface curvature for accurate triangular meshing. Pol Marit Res 2014; 21: 41–44. DOI: 10.2478/pomr-2014-0017.

53.

Gao

Hao

, et al. Grid generation on free-form surface using guide line advancing and surface flattening method. Adv Eng Software 2017; 110: 98–109. DOI: 10.1016/j.advengsoft.2017.04.003.

54.

Wang

Gao

, et al. A triangular mesh generator over free-form surfaces for architectural design. Autom ConStruct 2018; 93: 280–292. DOI: 10.1016/j.autcon.2018.05.018.

55.

Moore

. Understanding structures. New York, NY: McGraw-Hill Education, 1998.

An approach to automating the construction of 3D girihs (3D Islamic geometric patterns) on NURBS surfaces

Abstract

Keywords

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