Generalization of log-aesthetic curves via similarity geometry

Inoguchi, Jun-ichi; Ziatdinov, Rushan; Miura, Kenjiro T.

doi:10.1007/s13160-018-0335-7

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…Here we recall the notion of similarity curvature from [4,5]. Note that planar curves are determined by the similarity curvature uniquely up to similarity transformations.…”

Section: Preliminariesmentioning

confidence: 99%

“…One can see that the similarity curvature S is expressed by the Kummer confluent hypergeometric function and Tricomi confluent hypergeometric function (see [5] (p. 257)).…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, the location of the logarithmic spiral in the whole family is not clear. To overcome these difficulties, we use a new framework "similarity geometry" for the study of aesthetic curves developed in our previous works [4,5]. The usage of the log-aesthetic curve for practical design is still limited and we should extend its formula to obtain various curves to solve practical design problems, such as G n Hermite interpolation, deformation and smoothing; data-point fitting; and blending plural curves [5].…”

Section: Introductionmentioning

confidence: 99%

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A Note on Superspirals of Confluent Type

2020

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Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

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“…Here we recall the notion of similarity curvature from [4,5]. Note that planar curves are determined by the similarity curvature uniquely up to similarity transformations.…”

Section: Preliminariesmentioning

confidence: 99%

“…One can see that the similarity curvature S is expressed by the Kummer confluent hypergeometric function and Tricomi confluent hypergeometric function (see [5] (p. 257)).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on Superspirals of Confluent Type

2020

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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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“…The class of log-aesthetic curves has concerned some researchers, as these have many applications in industrial and graphical design. Various formulations of these curves have been studied by Inoguchi, Albayari, and Lu et al [25][26][27][28]. These curves usually have a non-polynomial form.…”

Section: Introductionmentioning

confidence: 99%

Synthesis of Two New Mechanisms Which Generate a Highly Aesthetic Design Image—The Flower of Life

et al. 2020

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(1) Background: The objective of this study was to develop the field of mechanisms that generate aesthetic curves, as they can be useful in designing machines for the manufacture of highly beautiful products. We started from the image of the Flower of Life and we set out to build mechanisms in the field of mechanical engineering that can generate this image. (2) Methods: First, we studied in detail the geometry of the figure. Secondly, based on the mathematical considerations, we made the synthesis of a mechanism able to trace this image, based on the Geneva mechanism with intermittent movement. We completed this mechanism with other kinematic chains, so we generated the proposed curve. Next, we made the synthesis of the second generating mechanism. This was a gear mechanism with compound profiles to which we added additional kinematic chains to generate the Flower of Life curve. Further on, kinematics was applied to confirm the drawing of the curves with the closed loop method. (3) Results: The results obtained following the study and presented in the paper show that the two created mechanisms generate images (trajectories) identical to the Flower of Life. Moreover, other images similar to the Flower of Life, with great artistic potential, were obtained. (4) Conclusions: The aim of the study was achieved, the authors proposed two new mechanisms which generate a highly aesthetic design image—the “Flower of Life”. The paper provides all the notions of structure and kinematics that are necessary to calculate the mechanisms and also analyzes other kinematic possibilities of the created mechanisms. The paper contributes to the development of a new field in the Theory of Mechanisms: generating aesthetic curves.

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Generalization of log-aesthetic curves via similarity geometry

Cited by 6 publications

References 17 publications

A Note on Superspirals of Confluent Type

A Note on Superspirals of Confluent Type

Symmetrical Patterns of Ainu Heritage and Their Virtual and Physical Prototyping

Synthesis of Two New Mechanisms Which Generate a Highly Aesthetic Design Image—The Flower of Life

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