Research

16 pages, 3642 KiB

Open AccessArticle

A Comparative Study of Different Schemes Based on Bézier-like Functions with an Application of Craniofacial Fractures Reconstruction

by

Abdul Majeed

,

Muhammad Abbas

and

Kenjiro T. Miura

Mathematics 2022, 10(8), 1269; https://doi.org/10.3390/math10081269 - 11 Apr 2022

Cited by 1 | Viewed by 1047

Abstract

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference [...] Read more.

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference skull, thin plate spline and radial basis functions have been used for cranial implant restoration. Computer Aided Geometric Design (CAGD) has also been used in bio-modeling and specifically for the restoration of cranial defects in form of different spline curves, namely

C^{1}, C^{2}, G C^{1} G C^{2},

rational curves, B-spline and Non-Uniform Rational B-Spline (NURBS) curves. This paper gives an in-depth comparison of existing techniques by highlighting the limitations and advantage in different contexts. The construction of craniofacial fractures is made using different Bézier-like functions (Ball, Bernstein and Timmer basis functions) and is analyzed in detail. The

C^{1}, G C^{1}

and

G C^{2}

cubic Ball curves are performed well for construction of the small fractured part. Any form of fracture is constructed using this approach and it has been effectively applied to frontal and parietal bone fractures. However, B-spline and NURBS curves can be used for any type of fractured parts and are more friendly user. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

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22 pages, 1597 KiB

Open AccessArticle

Construction of Cubic Trigonometric Curves with an Application of Curve Modelling

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Mathematics 2022, 10(7), 1087; https://doi.org/10.3390/math10071087 - 28 Mar 2022

Viewed by 1319

Abstract

This paper introduces new trigonometric basis functions (TBF) in polynomial and rational form with two shape parameters (SPs). Some classical characteristics, such as the partition of unity, positivity, symmetry, CHP, local control and invariance under affine transformation properties are proven mathematically and graphically. [...] Read more.

This paper introduces new trigonometric basis functions (TBF) in polynomial and rational form with two shape parameters (SPs). Some classical characteristics, such as the partition of unity, positivity, symmetry, CHP, local control and invariance under affine transformation properties are proven mathematically and graphically. In addition, different continuity conditions at uniform knots (UK) are proven. Some open and closed curves from TBS and trigonometric rational B-spline (TRBS) are generated to test the applicability of the suggested technique, and the influence of the shape parameter is also noted. Furthermore, various objects, such as designing an alphabet, star, butterfly, leaf and 3D cube. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

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14 pages, 958 KiB

Open AccessArticle

Curve and Surface Geometric Modeling via Generalized Bézier-like Model

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and

Mathematics 2022, 10(7), 1045; https://doi.org/10.3390/math10071045 - 24 Mar 2022

Cited by 1 | Viewed by 1335

Abstract

Generalized Bernstein-like functions (gB-like functions) with different shape parameters are used in this work. Parametric and geometric conditions in generalized form are developed. Some numerical examples of the parametric continuity (PC) and geometric continuity (GC) constraints of generalized Bézier-like curves (gB-like curves) are [...] Read more.

Generalized Bernstein-like functions (gB-like functions) with different shape parameters are used in this work. Parametric and geometric conditions in generalized form are developed. Some numerical examples of the parametric continuity (PC) and geometric continuity (GC) constraints of generalized Bézier-like curves (gB-like curves) are analyzed with graphical representation. Bézier-like symmetric rotation surfaces are constructed by gB-like curves. Vase and Capsule Taurus surfaces are modeled with the help of symmetry. The effect of shape parameters on surfaces are also analyzed. The illustrating figures reveal that the proposed curves and surfaces yield an accommodating strategy and mathematical depiction of Bézier curves and surfaces, allowing them to be a beneficial way to describe curves and surfaces. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

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15 pages, 1099 KiB

Open AccessArticle

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 1184

Abstract

We introduce a new sequence space

h_{A} (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

h_{A} (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, \dots)

, then

h_{A} (p)

reduces to the generalized Hahn space

h_{d}

. We applied our own software to visualize the shapes of parts of spheres in three-dimensional space endowed with the relative paranorm of

h_{A} (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

Open AccessArticle

A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters

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and

Mathematics 2022, 10(3), 376; https://doi.org/10.3390/math10030376 - 26 Jan 2022

Cited by 6 | Viewed by 1749

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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17 pages, 2803 KiB

Open AccessArticle

The Development of Log Aesthetic Patch and Its Projection onto the Plane

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and

Mathematics 2022, 10(1), 160; https://doi.org/10.3390/math10010160 - 05 Jan 2022

Viewed by 1371

Abstract

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify [...] Read more.

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

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14 pages, 7762 KiB

Open AccessArticle

Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation

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and

Mathematics 2021, 9(21), 2699; https://doi.org/10.3390/math9212699 - 24 Oct 2021

Cited by 2 | Viewed by 1596

Abstract

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) [...] Read more.

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by α. First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

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23 pages, 6033 KiB

Open AccessArticle

G³ Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications

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Mathematics 2021, 9(19), 2350; https://doi.org/10.3390/math9192350 - 22 Sep 2021

Cited by 4 | Viewed by 1326

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

G^{1}

,

G^{2}

(Farin-Boehm and Beta) and

G^{3}

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

Open AccessArticle

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

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Mathematics 2021, 9(18), 2212; https://doi.org/10.3390/math9182212 - 09 Sep 2021

Cited by 2 | Viewed by 1501

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

Open AccessArticle

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 - 02 Sep 2021

Cited by 15 | Viewed by 2634

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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