CONSURF. Part two: description of the algorithms

doi:10.1016/0010-4485(75)90068-8

Computer-Aided Design

Volume 7, Issue 4, October 1975, Pages 237-242

https://doi.org/10.1016/0010-4485(75)90068-8 Get rights and content

Abstract

The paper is the second of a series describing the surface lofting program CONSURF and outlines the algorithms within the program which transform the geometrical input into the mathematical variables of the conic lofting tile.

References (2)

A.A. Ball
CONSURF. Part one: introduction of the conic lofting tile
Comput. Aided Des.
(October 1974)
H. Akima
A new method of interpolation and smooth curve fitting based on local procedures
J. Assoc. for Comput. Mach.
(October 1970)

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Hybrid chameleon swarm algorithm with multi-strategy: A case study of degree reduction for disk Wang–Ball curves
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Cubic Ball curve was first proposed in 1974, and was created by British mathematician Ball as the mathematical basis of CONSURF fuselage surface modeling system [4–6].
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.
Combined cubic generalized ball surfaces: Construction and shape optimization using an enhanced JS algorithm
2023, Advances in Engineering Software
Citation Excerpt :
Ball curves and surfaces are widely used in geometric modeling fields such as computer-aided design and manufacturing, computer graphics, 3D medical imaging and computer animation. In 1974, mathematician Ball pioneered a class of cubic parameter curves [1–3], which are later called Ball curves. The traditional Ball curves cannot meet the requirements for the application of complex curves in industrial design and other fields because they are only three times.
The construction and shape optimization of complex shape-adjustable surfaces is a crucial and intractable technique in Computer Aided geometric Design (CAGD), which has a wide range of application value in many product designs and manufacturing fields involving complex surfaces modeling. In this paper, a novel combined cubic generalized Ball (CCG-Ball, for short) surfaces is constructed and the shape optimization of CCG-Ball surfaces is studied by an enhanced jellyfish search (JS) algorithm. First and foremost, we construct the CCG-Ball surfaces with multiple shape parameters based on a class of cubic generalized Ball basis functions, and then derive the conditions of G¹ and G² continuity for the surfaces. The shapes of CCG-Ball surfaces can be adjusted and optimized expediently by utilizing their shape parameters. Secondly, the shape optimization of CCG-Ball surfaces is mathematically an optimization problem that can be efficiently dealt with by swarm intelligence algorithm. In this regard, an enhanced JS termed EJS algorithm, combined with sine and cosine learning factors, local escape operator, opposition-based learning and quasi-opposition learning strategies, is introduced to improve the convergence speed and calculation accuracy of the JS algorithm. Finally, by minimizing the energy of CCG-Ball surfaces as the evaluation standard, the shape optimization models of the surfaces with G¹ and G² geometric continuity are established, respectively. The EJS algorithm is utilized to solve the established models, and the CCG-Ball surfaces with minimum energy are obtained. The example results illustrate the ability of EJS algorithm in effectively solving the shape optimization problems of complex CCG-Ball surfaces.
An enhanced manta ray foraging optimization algorithm for shape optimization of complex CCG-Ball curves
2022, Knowledge-Based Systems
Citation Excerpt :
Among them, Ball curves and surfaces constructed by Ball basis functions has been widely used and deeply studied because of their simple structure and excellent properties. Ball [2–4], a British mathematician, defined rational cubic Ball curves for the first time in 1974 and took the Ball curves as mathematical basis for CONSURF fuselage surface modeling system of Warton former British Airways. Because basis functions of rational cubic Ball curves are limited to cubic, many scholars further study generalized Ball curves of high degree.
The shape optimization of complex curves is a crucial and intractable technique in computer aided geometric design and widely used in many product design and manufacturing fields involving complex curve modeling. In this paper, an enhanced manta ray foraging optimization (MRFO) algorithm is used to optimize the shape of complex composite cubic generalized Ball (CCG-Ball, for short) curves. Firstly, to solve the problems of shape optimization for Ball curves, we construct a class of new cubic generalized Ball basis, and then present the CCG-Ball curves with multiple shape parameters based on the constructed basis functions. The shapes of the curves can be modified and optimized easily by using the shape parameters. Secondly, the shape optimization of CCG-Ball curves is mathematically an optimization problem that can be efficiently dealt with by swarm intelligence algorithm. In this regard, an enhanced MRFO called WMQIMRFO algorithm, combined with control parameter adjustment, wavelet mutation and quadratic interpolation strategy, is developed to enhance its capability of jumping out of the local minima and improve the calculation accuracy of the native algorithm. Furthermore, the superiority of the WMQIMRFO algorithm is verified by comparing with standard MRFO, other improved MRFO and popular nature-inspired optimization algorithms on the well-known CEC’14 and CEC’17 test suite as well as four engineering optimization problems, respectively. Finally, by minimizing the bending energy of the CCG-Ball curves as the evaluation standard, the shape optimization models of the curves with 1th-order and 2th-order geometric continuity are established, respectively. The WMQIMRFO algorithm is utilized to solve the established models, and the CCG-Ball curves with minimum energy are obtained. Some representative numerical examples illustrate the ability of the proposed WMQIMRFO algorithm in effectively solving the shape optimization problems of complex CCG-Ball curves in terms of precision, robustness, and convergence characteristics.
Bidiagonal decomposition of rectangular totally positive Said-Ball-Vandermonde matrices: Error analysis, perturbation theory and applications
2016, Linear Algebra and Its Applications
Citation Excerpt :
The Said–Ball basis is a generalization of the Ball basis [2–4], a well-known basis for cubic polynomials on a finite interval which is useful in the field of Computer-Aided Design.
An algorithm for computing the bidiagonal decomposition of Said–Ball–Vandermonde matrices in the rectangular case is presented, which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is fast and possesses high relative accuracy. The error analysis of the algorithm is also addressed, along with the perturbation theory for the bidiagonal factorization of totally positive Said–Ball–Vandermonde matrices. Some numerical experiments showing the good behavior of the algorithm are included.
Curve construction based on four αβ-Bernstein-like basis functions
2015, Journal of Computational and Applied Mathematics
Citation Excerpt :
Although the weights in the cubic non-uniform rational B-spline curves possess an effect for adjusting the shape of the curves [1–3], how to change the weights to adjust the shape of a curve is sometimes quite opaque to the user. In 1974, Ball used a kind of cubic basis to define his lofting surface program CONSURF [38–40]. This cubic basis is more efficient in evaluation than the cubic Bernstein basis.
Four new $α β$ -Bernstein-like basis functions with two exponential shape parameters, are constructed in this paper, which include the cubic Said–Ball basis functions and the cubic Bernstein basis functions. Within the general framework of Quasi Extended Chebyshev space, we prove that the proposed $α β$ -Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding $α β$ -Bézier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar $α β$ -Bézier-like curve having single or double inflection points, a loop or a cusp, or be locally or globally convex in terms of the relative position of its control polygons’ side vectors. Based on the new proposed $α β$ -Bernstein-like basis, a class of $α β$ -B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated $α β$ -B-spline-like curves have $C^{2}$ continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be $C^{2} \cap F C^{k + 3}$ ( $k \in Z^{+}$ ) continuous for particular choice of shape parameters. The exponential shape parameters serve as tension shape parameters and play a predictable adjusting role on generating curves.
Ameliorated Snake Optimizer-Based Approximate Merging of Disk Wang–Ball Curves
2024, Biomimetics

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CONSURF. Part two: description of the algorithms

Abstract

Comput. Aided Des.

A new method of interpolation and smooth curve fitting based on local procedures

J. Assoc. for Comput. Mach.