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Mathematics
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Modern Geometric Modeling: Theory and Applications II
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Modern Geometric Modeling: Theory and Applications II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 16747

Special Issue Editors

Prof. Dr. Kenjiro T. Miura

E-Mail Website1 Website2
Guest Editor
Department of Mechanical Engineering, Shizuoka University, Hamamatsu, Japan
Interests: geometric modeling; aesthetic curves and surfaces; image processing; intelligent optical measurement; computing
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Rushan Ziatdinov

E-Mail Website1 Website2
Guest Editor
Department of Industrial Engineering, Keimyung University, Daegu 704-701, Korea
Interests: geometric modeling; computer-aided geometric design; computer-aided design; scientific visualization; computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent decades, geometric modeling has evolved into an interesting and powerful branch of modern science and engineering. Its theories are mostly related to mathematics and computer science, and applications are commonly found in industrial design, computer graphics and animation, CAD/CAM, architecture, and other areas. Most of the popular approaches in geometric modeling include parametric spline curves and surfaces (NURBS, B-splines, T-splines, etc.) which are simple and intuitive for use by industrial and graphic designers. On the other hand, high-quality shapes often require non-traditional approaches such as the use of special functions.

We believe that the field of geometric modeling needs breakthrough research which will result in a higher level of understanding of shape modeling and visual perception, geometric aesthetics, the need of artificial intelligence and the multi-criteria assessment of shape quality in the CAD systems of the future, as well as the necessity of fundamentally new mathematical tools and paradigms which will revolutionize geometric modeling.

The scope of the Special Issue includes but is not limited to original research works within the subject of geometric modeling and its applications in engineering, arts, physics, biology, medicine, computer graphics, architecture, etc., as well as theoretical mathematics and geometry which can be applied to problems of geometric modeling. For this Special Issue, we plan to accept the following types of manuscripts:

  1. Overviews;
  2. Research manuscripts;
  3. Short manuscripts which discuss open problems in geometric modeling.

In view of the above, we invite you to submit your latest work to this Special Issue entitled “Modern Geometric Modeling: Theory and Applications II”. If you are interested in the contents of our previous Special Issue, you can access it online at https://www.mdpi.com/journal/mathematics/special_issues/Modern_Geometric_Modeling_Theory_Applications 

Prof. Dr. Kenjiro T. MIURA
Prof. Dr. Rushan Ziatdinov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • curve, surface, and solid modeling
  • mathematical design
  • geometric modeling in arts
  • special functions in geometric modeling
  • applied, discrete, and computational geometry and topology
  • isogeometric analysis high-quality curves and surfaces
  • non-polynomial curves and surfaces (spirals, log-aesthetic curves, GLACS, superspirals, quaternion curves, etc.)
  • mesh generation
  • three-dimensional optical illusions
  • industrial and scientific applications

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Published Papers (5 papers)

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Research

15 pages, 1099 KiB  
Article
Modeling Spheres in Some Paranormed Sequence Spaces
by Vesna I. Veličković, Eberhard Malkowsky and Edin Dolićanin
Mathematics 2022, 10(6), 917; https://doi.org/10.3390/math10060917 - 13 Mar 2022
Cited by 2 | Viewed by 1701
Abstract
We introduce a new sequence space hA(p), which is not normable, in general, and show that it is a paranormed space. Here, A and p denote an infinite matrix and a sequence of positive numbers. In the special [...] Read more.
We introduce a new sequence space hA(p), which is not normable, in general, and show that it is a paranormed space. Here, A and p denote an infinite matrix and a sequence of positive numbers. In the special case, when A is a diagonal matrix with a sequence d of positive terms on its diagonal and p=(1,1,…), then hA(p) reduces to the generalized Hahn space hd. We applied our own software to visualize the shapes of parts of spheres in three-dimensional space endowed with the relative paranorm of hA(p), when A is an upper triangle. For this, we developed a parametric representation of these spheres and solved the visibility and contour (silhouette) problems. Finally, we demonstrate the effects of the change of the entries of the upper triangle A and the terms of the sequence p on the shape of the spheres. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)
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19 pages, 3134 KiB  
Article
A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters
by Moavia Ameer, Muhammad Abbas, Thabet Abdeljawad and Tahir Nazir
Mathematics 2022, 10(3), 376; https://doi.org/10.3390/math10030376 - 26 Jan 2022
Cited by 9 | Viewed by 2698
Abstract
Bézier curves and surfaces with shape parameters have received more attention in the field of engineering and technology in recent years because of their useful geometric properties as compared to classical Bézier curves, as well as traditional Bernstein basis functions. In this study, [...] Read more.
Bézier curves and surfaces with shape parameters have received more attention in the field of engineering and technology in recent years because of their useful geometric properties as compared to classical Bézier curves, as well as traditional Bernstein basis functions. In this study, the generalized Bézier-like curves (gBC) are constructed based on new generalized Bernstein-like basis functions (gBBF) with two shape parameters. The geometric properties of both gBBF and gBC are studied, and it is found that they are similar to the classical Bernstein basis and Bézier curve, respectively. Some free form curves can be modeled using the proposed gBC and surfaces as the applications. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)
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23 pages, 6033 KiB  
Article
G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications
by Samia BiBi, Md Yushalify Misro, Muhammad Abbas, Abdul Majeed and Tahir Nazir
Mathematics 2021, 9(19), 2350; https://doi.org/10.3390/math9192350 - 22 Sep 2021
Cited by 4 | Viewed by 1971
Abstract
In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points [...] Read more.
In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points and planes with GHT-Bézier curve. With different shape control parameters in their domain, a class of GHT-Bézier developable surfaces can be established (such as enveloping developable GHT-Bézier surfaces, spine curve developable GHT-Bézier surfaces, geodesic interpolating surfaces for GHT-Bézier surface and developable GHT-Bézier canal surfaces), which possess many properties of GHT-Bézier surfaces. By changing the values of shape parameters the effect on the developable surface is obvious. In addition, some useful geometric properties of GHT-Bézier developable surface and the G1, G2 (Farin-Boehm and Beta) and G3 continuity conditions between any two GHT-Bézier developable surfaces are derived. Furthermore, various useful and representative numerical examples demonstrate the convenience and efficiency of the proposed method. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)
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20 pages, 20323 KiB  
Article
Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm
by Xiaomin Liu, Muhammad Abbas, Gang Hu and Samia BiBi
Mathematics 2021, 9(18), 2212; https://doi.org/10.3390/math9182212 - 9 Sep 2021
Cited by 2 | Viewed by 2264
Abstract
Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We [...] Read more.
Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)
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32 pages, 3836 KiB  
Article
Generalized Fractional Bézier Curve with Shape Parameters
by Syed Ahmad Aidil Adha Said Mad Zain, Md Yushalify Misro and Kenjiro T. Miura
Mathematics 2021, 9(17), 2141; https://doi.org/10.3390/math9172141 - 2 Sep 2021
Cited by 21 | Viewed by 3799
Abstract
The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how [...] Read more.
The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)
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