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25 pages, 1928 KiB

Open AccessArticle

A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

by Yunyi Fu and Yuanpeng Zhu

Mathematics 2021, 9(10), 1154; https://doi.org/10.3390/math9101154 - 20 May 2021

Cited by 1 | Viewed by 1904

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions

α (t)

and

β (t)

are constructed in a generalized quasi cubic trigonometric space span [...] Read more.

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions

α (t)

and

β (t)

are constructed in a generalized quasi cubic trigonometric space span

{1, \sin^{2} t, {(1 - \sin t)}^{2} α (t), {(1 - \cos t)}^{2} β (t)}

, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions

α_{i} (t)

and

β_{i} (t)

is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and

C^{2}

continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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13 pages, 2732 KiB

Open AccessFeature PaperArticle

Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning

by Kinga Kruppa, Roland Kunkli and Miklós Hoffmann

Mathematics 2021, 9(8), 843; https://doi.org/10.3390/math9080843 - 13 Apr 2021

Cited by 1 | Viewed by 1651

Abstract

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the

R^{2, 1}

Minkowski space. Extending the class of MPH curves, a new class [...] Read more.

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the

R^{2, 1}

Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These are special curves in

R^{2, 1}

that define rational boundaries for the corresponding domain. A method to use RE and MPH curves for skinning purposes, i.e., for circle-based modeling, has been developed recently. In this paper, we continue this study by proposing a new, more flexible way how these curves can be used for skinning a discrete set of circles. We give a thorough overview of our algorithm, and we show a significant advantage of using RE and MPH curves for skinning purposes: as opposed to traditional skinning methods, unintended intersections can be detected and eliminated efficiently. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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11 pages, 944 KiB

Open AccessArticle

A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

by Antonio Falcó, Lucía Hilario, Nicolás Montés, Marta C. Mora and Enrique Nadal

Mathematics 2020, 8(12), 2245; https://doi.org/10.3390/math8122245 - 19 Dec 2020

Cited by 9 | Viewed by 3226

Abstract

A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem [...] Read more.

A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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19 pages, 1414 KiB

Open AccessArticle

Neural-Network-Based Curve Fitting Using Totally Positive Rational Bases

by Rocio Gonzalez-Diaz, E. Mainar, Eduardo Paluzo-Hidalgo and B. Rubio

Mathematics 2020, 8(12), 2197; https://doi.org/10.3390/math8122197 - 10 Dec 2020

Cited by 3 | Viewed by 3128

Abstract

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural [...] Read more.

This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational Bézier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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10 pages, 2919 KiB

Open AccessArticle

Further Properties of Quantum Spline Spaces

by Gülter Budakçı and Halil Oruç

Mathematics 2020, 8(10), 1770; https://doi.org/10.3390/math8101770 - 14 Oct 2020

Cited by 3 | Viewed by 1587

Abstract

We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for

1 / q

-integration and [...] Read more.

We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for

1 / q

-integration and

1 / q

-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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15 pages, 3097 KiB

Open AccessArticle

An Improvement on the Upper Bounds of the Partial Derivatives of NURBS Surfaces

by Ye Tian, Tao Ning, Jixing Li, Jianmin Zheng and Zhitong Chen

Mathematics 2020, 8(8), 1382; https://doi.org/10.3390/math8081382 - 18 Aug 2020

Viewed by 1873

Abstract

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order [...] Read more.

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order partial derivatives of NURBS surfaces. A pilot study was performed using inequality theorems and degree reduction of B-spline basis functions. Theoretical analysis provides simple forms of the new bounds. Numerical examples are performed to illustrate that our method has sharper bounds than the existing ones. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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17 pages, 1203 KiB

Open AccessArticle

Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

by Jinhua Qian, Xueqian Tian and Young Ho Kim

Mathematics 2020, 8(6), 919; https://doi.org/10.3390/math8060919 - 5 Jun 2020

Cited by 1 | Viewed by 1853

Abstract

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal [...] Read more.

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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15 pages, 306 KiB

Open AccessArticle

Geometric Properties and Algorithms for Rational q-Bézier Curves and Surfaces

by Jorge Delgado and J. M. Peña

Mathematics 2020, 8(4), 541; https://doi.org/10.3390/math8040541 - 7 Apr 2020

Cited by 10 | Viewed by 2676

Abstract

In this paper, properties and algorithms of q-Bézier curves and surfaces are analyzed. It is proven that the only q-Bézier and rational q-Bézier curves satisfying the boundary tangent property are the Bézier and rational Bézier curves, respectively. Evaluation algorithms formed by steps in [...] Read more.

In this paper, properties and algorithms of q-Bézier curves and surfaces are analyzed. It is proven that the only q-Bézier and rational q-Bézier curves satisfying the boundary tangent property are the Bézier and rational Bézier curves, respectively. Evaluation algorithms formed by steps in barycentric form for rational q-Bézier curves and surfaces are provided. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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