Generalization of log-aesthetic curves by Hamiltonian formalism

Satō, Masayuki; Shimizu, Yasuhiro

doi:10.14495/jsiaml.8.49

Cited by 7 publications

(3 citation statements)

References 10 publications

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“…Though the presented methodology worked well for the motifs shown in this article, other quantitative analysis of the motifs can be performed to understand their aesthetic nature in a quantitative sense. As such, the methodology can be enriched by incorporating other faculties of thought regarding the free-form curve, e.g., log-aesthetic curve [29,30]. In addition, the data acquisition step of the presented methodology (i.e., the first half of the modeling domain shown in Figure 2) relies on human perception only.…”

Section: Virtual and Physical Prototypes Of Ainu Patternsmentioning

confidence: 99%

Symmetrical Patterns of Ainu Heritage and Their Virtual and Physical Prototyping

Tashi

Ullah

2019

Symmetry

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This article addresses virtual and physical prototyping of some symmetrical patterns collected from the Ainu cultural heritage. The indigenous people living in the northern part of Japan (e.g., Hokkaido), known as Ainu, often decorate their houses, clothing, ornaments, utensils, and spiritual goods using some unique patterns. The patterns carry their identity as well as their sense of aesthetics. Nowadays, different kinds of souvenirs and cultural artifacts crafted with Ainu patterns are cherished by many individuals in Japan and abroad. Thus, the Ainu patterns carry both cultural and commercial significance. A great deal of craftsmanship is needed to produce the Ainu patterns precisely. There is a lack of human resources having such craftsmanship. It will remain the same in the foreseeable future. Thus, there is a pressing need to preserve such craftsmanship. Digital manufacturing technology can be used to preserve the Ainu pattern-making craftsmanship. From this perspective, this article presents a methodology to create both virtual and physical prototypes of Ainu patterns using digital manufacturing technology. In particular, a point cloud-based approach was adopted to model the patterns. A point cloud representing a pattern was then used to create a virtual prototype of the pattern in the form of a solid CAD model. The triangulation data of each solid CAD model were then used to run a 3D printer to produce a physical prototype (replica of the pattern). The virtual and physical prototypes of both basic (Hokkaido) Ainu motifs and some synthesized patterns were reproduced using the presented methodology. The findings of this study will help those who want to digitize the craftsmanship of culturally significant artifacts without using a 3D scanner or image processing.

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Section: Virtual and Physical Prototypes Of Ainu Patternsmentioning

confidence: 99%

Symmetrical Patterns of Ainu Heritage and Their Virtual and Physical Prototyping

Tashi

Ullah

2019

Symmetry

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“…For instance, the slope α is expressed as α = 1 + u ′ /u 2 . LAC and qAC are reformulated in terms of similarity geometry as follows [6,17,18]: A plane curve γ(θ) is said to be a LAC of slope α if its similarity curvature satisfies the Bernoulli equation:…”

Section: Log-aesthetic Curves and Similarity Geometrymentioning

confidence: 99%

Log-aesthetic curves as similarity geometric analogue of Euler's elasticae

Inoguchi

Kajiwara

Miura

et al. 2018

Computer Aided Geometric Design

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In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler-Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.

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“…Based on this reformulation, qAC is introduced in the following manner [11]. A plane curve in similarity geometry is said to be a qAC of slope α if its similarity curvature obeys the Riccati equation du dθ…”

Section: Log-aesthetic Curves and Similarity Geometrymentioning

confidence: 99%

Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles

Inoguchi,

Jikumaru,

Kajiwara

et al. 2018

Preprint

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In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which is used in CAGD. We consider these curves in the context of similarity geometry and characterize them in terms of a "stationary" integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formalism developed here, we propose a discretization of both the curves and the associated variational principle which preserves the underlying integrable structure. We finally present an algorithm for the generation of discrete log-aesthetic curves for given G 1 data. The computation time to generate discrete log-aesthetic curves is much shorter than that for numerical discretizations of log-aesthetic curves due to the avoidance of fine numerical integration to calculate their shapes. Instead, only coarse summation is required.

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Generalization of log-aesthetic curves by Hamiltonian formalism

Cited by 7 publications

References 10 publications

Symmetrical Patterns of Ainu Heritage and Their Virtual and Physical Prototyping

Symmetrical Patterns of Ainu Heritage and Their Virtual and Physical Prototyping

Log-aesthetic curves as similarity geometric analogue of Euler's elasticae

Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles

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