Automatic generation of hyperbolic drawings

Ouyang, Peichang; Fathauer, Robert; Chung, Kwok Wai; Wang, Xinchang

doi:10.1016/j.amc.2018.09.052

Cited by 8 publications

(7 citation statements)

References 26 publications

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“…Substituting ( 14), (23), and ( 24) into (25) and noting that there is only one unknown 𝛿 1 , we solve the equation for tan 𝛿 1 and obtain (20) and (21).…”

Section: Transformations For Constructing Mhstsmentioning

confidence: 99%

“…Based on the principles of symmetry, Escher created plenty of popular and fascinating artworks and had a durable and profound influence in the normally disparate fields of art and mathematics [7][8][9]. Interestingly, due to the aesthetic attraction as well as commercial potential, there appeared a lot of studies dedicated to the creation of Escher-like artworks, such as Escherization [10][11][12], 3D Escher-like tilings [13,14], metamorphosis [15,16], Escher transmutation [17,18], fractal [19], and hyperbolic drawings [20,21]. The most outstanding feature of the resulting images is that motifs are easily recognizable, which presents a true and natural artistic flavor.…”

Section: Introductionmentioning

confidence: 99%

“…At the same time, Circle Limit I-IV have almost become the most popular and indispensable material for science popularization of hyperbolic geometry. Not surprisingly, such drawings have motivated much research for creating hyperbolic patterns and have inspired and promoted much work aimed at producing artworks using computer graphics [20,21,31]. It is worth noting that there is nothing in the literature concerning spiral drawings in hyperbolic geometry.…”

Section: Introductionmentioning

confidence: 99%

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Generation of Escher‐like spiral drawings in a modified hyperbolic space

Chung

Ouyang

Nicolas³

et al. 2023

Math Methods in App Sciences

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Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral drawings in hyperbolic space. In this paper, we consider a modified hyperbolic geometry by removing the condition that a geodesic is orthogonal to the unit circle in the Poincaré model. We show that spiral symmetry and the similarity property exist in this modified geometry so that the creation of uncommon hyperbolic spiral drawings is possible. To this end, we first establish the theoretical foundation for the proposed method by deriving a contraction mapping and a rotation for constructing modified hyperbolic spiral tilings (MHSTs) and introduce symmetry groups to analyze the structure of MHSTs. Then, to embed a pre‐designed wallpaper template into the tiles, we derive a one‐to‐one mapping between a tile of MHST and a rectangle. Finally, we specify some technical implementation details and give a gallery of the resulting MHST drawings. Using existing wallpaper templates, the proposed method is able to generate a great variety of exotic Escher‐like drawings.

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“…Substituting ( 14), (23), and ( 24) into (25) and noting that there is only one unknown 𝛿 1 , we solve the equation for tan 𝛿 1 and obtain (20) and (21).…”

Section: Transformations For Constructing Mhstsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Chung

Ouyang

Nicolas³

et al. 2023

Math Methods in App Sciences

Self Cite

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“…Furthermore, other non-Euclidean geometries are used in the generation of patterns. For instance, in [7], the authors used spherical geometry, whereas in [8], hyperbolic geometry was used. In the generation of patterns, not only various types of non-Euclidean geometries are used.…”

Section: Introductionmentioning

confidence: 99%

Procedural Generation of Artistic Patterns Using a Modified Orbit Trap Method

Gdawiec

Adewinbi

2022

Applied Sciences

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In the literature, we can find various methods for generating artistic patterns. One of the methods is the orbit trap method. In this paper, we propose various modifications of a variant of the orbit trap method that generates patterns with wallpaper symmetry. The first modification relies on replacing the Picard iteration (used in the original method) with the S-iteration known from the fixed point theory. Moreover, we extend the parameters in the S-iteration from scalar to vector ones. In the second modification, we replace the Euclidean metric used in the orbit traps with other metrics. Finally, we propose three new orbit traps. The presented examples show that using the proposed method, we are able to obtain a great variety of interesting patterns. Moreover, we show that a proper selection of the orbit traps and the mapping used by the method can lead to patterns that possess a local fractal structure.

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“…'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean" [15]. As people pay more and more attention to mathematics, art, aesthetics, and intelligence education, Escher's artworks have attracted more and more mathematical attention, building on his legacy with tools not available in his day, specifically of the computer: Escherization [19], 3D Escher-like tessellations [16,36], Metamorphosis [17], Escher transmutation [23,31], hyperbolic tessellations [26,32,34], and f -tilings [25]. On the other hand, a common shortcoming of computergenerated patterns to laypeople is that they lack interest, of an abstract nature, and so are thus of less obvious appeal [21,24,29], whereas with a recognisable motif, as Escherlike, there is thus an obvious point of interest.…”

mentioning

confidence: 99%

Generation of advanced Escher-like spiral tessellations

et al. 2021

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In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.

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Automatic generation of hyperbolic drawings

Cited by 8 publications

References 26 publications

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Procedural Generation of Artistic Patterns Using a Modified Orbit Trap Method

Generation of advanced Escher-like spiral tessellations

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