Disk Bézier curves

doi:10.1016/S0167-8396(98)00016-8

Computer Aided Geometric Design

Volume 15, Issue 7, July 1998, Pages 721-737

https://doi.org/10.1016/S0167-8396(98)00016-8 Get rights and content

Abstract

Disk Bézier curves are defined. Properties of these curves discussed are algorithms, a centered form, their use in approximating a real curve, their envelope and an approximation to the curves. Ball Bézier curves in three dimensions are introduced and some comments on the generalization of the results for disk Bézier curves to ball Bézier curves are given.

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There are more references available in the full text version of this article.

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Disk B-spline curve provides an alternative shape modeling tool to standard control point based modeling techniques by extending control points to control disks, resulting in an easy and straightforward way of modeling regions or shapes with adjustable thickness. Due to this feature, disk B-spline curves are frequently applied in artistic shape modeling applications, such as brushstroke representation, calligraphy, and animation, but they are also suitable for engineering purposes. However, the boundary curves of the described shapes tend to have unintentional self-intersections or cusps, negatively affecting the usability and artistic value of the resulting shape and texture. In this paper, we introduce an iterative algorithm to give a solution to this problematic issue with the help of an approximating circle skinning method, while preserving the advantageous properties of the disk B-spline curves. Our method also results in smoother texturing of the shape around the more sharply curved sections.
Hybrid chameleon swarm algorithm with multi-strategy: A case study of degree reduction for disk Wang–Ball curves
2023, Mathematics and Computers in Simulation
Citation Excerpt :
On this basis, Tuohy et al. [68] used interval B-spline curve and surface to approximate the measurement data points, and Lin et al. [43] studied the boundary evaluation algorithm of interval Bezier curve in 2D and 3D. Considering that the intervals are not rotationally symmetric in 2D plane and will continue to expand during the calculation process [12], Lin et al. [44] proposed disk Bézier curve by replacing the control vertex with a disk, and discussed some basic properties and boundary representations of the disk Bézier curve. Subsequently, Chen et al. [13] studied the whole set of algorithms for the isometric approximation of disk Bézier curve, Shi et al. [64] generalized the rational disk Bézier curve, Seah et al. [61] proposed a disk B-spline curve and applied it to artistic strokes and 2D animations.
In this paper, an enhanced hybrid chameleon swarm algorithm (CSA) is proposed and applied to the degree reduction problem of disk Wang–Ball (DWB) curve. CSA is a novel population-based algorithm inspired by the hunting behavior of chameleons, its simplicity and easy implementation make it applied to different fields. However, it suffers from premature convergence and easy to fall into local optimum, especially in the face of complex optimization problems. Therefore, this paper proposes an enhanced hybrid CSA (CCECSA, for short). Compared with the classic CSA, the proposed CCECSA mainly introduces three improvements: (1) The crisscross optimization algorithm is mixed to avoid premature convergence, in which the horizontal and vertical crossover can generate moderation solutions to increase the diversity of the population. (2) Elite guidance mechanism is introduced to speed up the convergence. (3) Competitive substitution mechanism is added to replace the worst individual, and an interference strategy is set to prevent the algorithm from falling into a local optimum. The efficiency and robustness of the proposed CCECSA are demonstrated by the comparison results with some advanced meta-heuristic algorithms on CEC2014, CEC2017, and 4 engineering design examples. In addition, for the degree reduction problem of DWB curves, the multi-degree reduction optimization models of its center curve and radius function are established respectively. At the same time, the optimal center curve and radius function of the approximating DWB curves of lower degree are obtained by the proposed CCECSA. The experimental results show that the proposed CCECSA achieves the optimal solution with better convergence and robustness. The source code of CCECSA is publicly available in the supplementary material related to this article.
Ball Said-Ball curve: Construction and its geometric algorithms
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Compared with the traditional Bézier curves, the control points of the interval curve are enclosed in a box, which is called control interval. Lin et al. proposed a more natural extension method by considering the control points as a set consisting of all points within a fixed distance of a given point, which is called control disk [10]. If the Bézier curves are computed using control disks then they will be called disk Bézier curves.
The representation method of curves and surfaces is an essential tool in industrial design, while a ball curve with adjustable thickness has its advantages in representing freedom tubular objects. By combing the Said-Ball basic functions and the control balls, this paper constructs the ball Said-Ball (BSB, for short) curve and discusses the related geometrical properties and algorithms. Then, the G¹ and G² continuity conditions between two adjacent BSB curves are discussed. Furthermore, on the premise of G² continuity, an extension method of the BSB curve with minimum energy is studied. For this method, the GaussNewton-NL2SOL algorithm is used to obtain the best scale parameters of the center curve. Meanwhile, the optimal parameters of the radius function are also determined by solving the extreme. Finally, the construction of the BSB rotation surface with different continuity is given. The provided modeling examples show that the BSB curve with thickness is more suitable and valuable in the design for the modeling of objects, on the premise of maintaining excellent geometric properties.
An intersection algorithm for disk B-spline curves
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In our work, we implement our algorithm for the DBSC of degree 3. The idea of the clipping method is originated from the well-known Bézier clipping method [29]. As we all know, Bézier clipping is an efficient method to quickly exclude regions that have no intersections.
In this paper, we propose a method for finding all 2D intersection regions between disk B-spline curves (DBSCs), which is very crucial for DBSC’s wide applications such as computer calligraphy, computer 2D animations, and non-photorealistic rendering. As a DBSC represents a region on a plane, the intersection of two DBSCs is a 2D region. To determine the region, the key is to compute the intersection points of the boundaries of two DBSCs. In our algorithm, the boundary of a DBSC is decomposed into four components: the upper boundary, the lower boundary, the start arc, and the end arc. The intersection of two DBSCs can be converted into the intersections between these four components. The main difficulty is to find the intersection involving the upper and lower boundaries of the two DBSCs, as they are variable offsets from the skeletons of the DBSC that are B-spline curves. In our approach, first the DBSCs are subdivided into several disk Bézier curves (DBCs). Therefore the problem of computing intersections of the DBSCs is converted into computing intersection of two DBCs. Then, the disk Bézier clipping method is proposed to exclude regions that have no intersection for the intersection of the two DBCs. In the case of where there is an intersection, we calculate the comparatively rough intersection to be used as initial values for later refinement through the disk Bézier clipping method. Besides, high precision (up to 10e $-$ 15) intersections are achieved by using the Newton’s iteration, which is quadratic convergent. The experimental results demonstrate that our algorithm can very efficiently compute all intersections between DBSCs with high precision. Our main contributions in this paper are as follows. First, for the first time, we give the direct parametric expression of DBSC’s boundary, which can be simply and conveniently used to compute the properties of DBSC’s boundary. Second, our proposed approach of calculating high-accuracy intersections of DBSCs makes DBSC a flexible and effective stroke representation that can be applied to further research such as corresponding computation, automatically coloring, region detection and so on.
Ameliorated Snake Optimizer-Based Approximate Merging of Disk Wang–Ball Curves
2024, Biomimetics
Dynamic Ball B-Spline Curves
2024, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

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Disk Bézier curves

Abstract

Curves and Surfaces in Computer Aided Geometric Design

Computational Geometry for Design and Manufacture

Ein Versuch zur Verallgemeinerung der Intervallrechnung

Computational geometry and software engineering: towards a geometric computing environment

Circular arithmetic and the determination of polynomial zeros

Numer. Math.