Computer-aided Geometric Design Research Articles

Dupin Cyclides Passing through a Fixed Circle

Dupin cyclides are classical algebraic surfaces of low degree. Recently, they have gained popularity in computer-aided geometric design (CAGD) and architecture owing to the fact that they contain many circles. We derive algebraic conditions that fully characterize the Dupin cyclides passing through a fixed circle. The results are applied to the basic problem in CAGD of the blending of Dupin cyclides along circles.

On Intersections of B-Spline Curves

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods.

An Optimal Property of B-Bases for the Modified Richardson Method

A space with a normalized totally positive basis has a unique normalized B-basis. In computer-aided geometric design, normalized B-bases present optimal shape-preserving properties. More optimal properties of normalized B-bases were proved previously. This paper provides a new optimal property concerning the modified Richardson iterative method when applied to collocation matrices of normalized B-bases. Moreover, a similar optimal property is proved for the tensor product of normalized B-bases.

Ameliorated Snake Optimizer-Based Approximate Merging of Disk Wang-Ball Curves.

A method for the approximate merging of disk Wang-Ball (DWB) curves based on the modified snake optimizer (BEESO) is proposed in this paper to address the problem of difficulties in the merging of DWB curves. By extending the approximate merging problem for traditional curves to disk curves and viewing it as an optimization problem, an approximate merging model is established to minimize the merging error through an error formulation. Considering the complexity of the model built, a BEESO with better convergence accuracy and convergence speed is introduced, which combines the snake optimizer (SO) and three strategies including bi-directional search, evolutionary population dynamics, and elite opposition-based learning. The merging results and merging errors of numerical examples demonstrate that BEESO is effective in solving approximate merging models, and it provides a new method for the compression and transfer of product shape data in Computer-Aided Geometric Design.

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space.

In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-[Formula: see text]. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-[Formula: see text]. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space-[Formula: see text] to demonstrate the applicability of the technique in Minkowski space.

The Quadratic Trigonometric Bezier like Curve With two Shape Parameters

The quadratic trigonometric Bezier model serves as a potent and valuable tool in computer-aided geometric design, boasting exceptional geometric properties. In this study, Novel functions are introduced by leveraging a trigonometric Bezier-like curve with adjustable shape parameters. Each curve segment is formed by three consecutive control points, providing precise control over shape by adjusting these parameters while keeping the control polygon unchanged. These curves exhibit superior proximity to the control polygon compared to quadratic Bezier curves across all shape parameter values, gradually approaching the control polygon as the shape parameter increases. Moreover, for practical application, the quadratic trigonometric Bezier-like curve was utilized to artfully design the Arabic word "الله" (Allah) and define flower contours, employing MATHEMATICA software. The results underscore the model's effectiveness in accurately representing intricate shapes, offering a versatile and efficient solution for diverse design challenges. Furthermore, the fusion of the quadratic trigonometric Bezier-like curve with shape parameters significantly augments design flexibility and adaptability, making it a valuable asset in computer graphics, geometric modeling, and artistic design across multiple domains.

The quasi-frame of the rational and polynomial Bezier curve by algorithm method in Euclidean space

PurposeFrames play an important role in determining the geometric properties of the curves such as curvature and torsion. In particular, the determination of the point types of the curve, convexity or concavity is also possible with the frames. The Serret-Frenet frames are generally used in curve theory. However, the Serret-Frenet frame does not work when the second derivative is zero. In order to eliminate this problem, the quasi-frame was obtained. In this study, the quasi frames of the polynomial and rational Bezier curves are calculated by an algorithmic method. Thus, it will be possible to construct the frame even at singular points due to the second derivative of the curve. In this respect, the contribution of this study to computer-aided geometric design studies is quite high.Design/methodology/approachIn this study, the quasi frame which is an alternative for all intermediate points of the rational Bezier curves was generated by the algorithm method, and some variants of this frame were analyzed. Even at the points where the second derivative of such rational Bezier curves is zero, there is a curve frame.FindingsSeveral examples presented at the end of the paper regarding the quasi-frame of the rational Bezier curve, polynomial Bezier curve, linear, quadratic and cubic Bezier curves emphasize the efficacy and preciseness.Originality/valueThe quasi-frame of a rational Bezier curve is first computed. Owing to the quasi frame, it will have been found a solution for the nonsense rotation of the curve around the tangent.

Fairing of planar curves to log-aesthetic curves

We present an algorithm to fair a given planar curve by a log-aesthetic curve (LAC). We show how a general LAC segment can be uniquely characterized by seven parameters and present a method of parametric approximation based on this fact. This work aims to provide tools to be used in reverse engineering for computer-aided geometric design. Finally, we show an example of usage by applying this algorithm to the data points obtained from 3D scanning a model-car roof.

Application of Orthogonal Polynomial in Orthogonal Projection of Algebraic Surface

Point orthogonal projection onto an algebraic surface is a very important topic in computer-aided geometric design and other fields. However, implementing this method is currently extremely challenging and difficult because it is difficult to achieve to desired degree of robustness. Therefore, we construct an orthogonal polynomial, which is the ninth formula, after the inner product of the eighth formula itself. Additionally, we use the Newton iterative method for the iteration. In order to ensure maximum convergence, two techniques are used before the Newton iteration: (1) Newton’s gradient descent method, which is used to make the initial iteration point fall on the algebraic surface, and (2) computation of the foot-point and moving the iterative point to the close position of the orthogonal projection point of the algebraic surface. Theoretical analysis and experimental results show that the proposed algorithm can accurately, efficiently, and robustly converge to the orthogonal projection point for test points in different spatial positions.

Construction of Cubic Bezier Curve

The construction of Bezier curves is one of the curves that are commonly discussed in Computer-Aided Geometric Design (CAGD). This study focuses on cubic Bezier curve. The objectives in this study are to review the properties of cubic Bezier curve and construct the cubic Bezier curves. In this study, the expanding equations from the basis function of the curve is used to construct the cubic Bezier curve. Future researchers can expand the degree of the Bezier curves, which is more useful in Computer-Aided Design (CAD), CAGD and engineering. The next studies in Bezier curve are recommended as a contribution for further research.

A Convolution-Based Computational Technique for Subdivision Depth of Doo-Sabin Subdivision Surface

Subdivision surface schemes are used to produce smooth shapes, which are applied for modelling in computer-aided geometric design. In this paper, a new and efficient numerical technique is presented to estimate the error bound and subdivision depth of the uniform Doo-Sabin subdivision scheme. In this technique, first, a result for computing bounds between P k (a polygon at k th level) and P ∞ (limit surface) of the Doo-Sabin scheme is obtained. After this, subdivision depth (the number of iterations) is computed by using the user-defined error tolerance. In addition, the results of the proposed technique are verified by taking distinct valence numbers of the Doo-Sabin surface scheme.

Shape optimisation of stainless steel corrugated cylindrical shells for additive manufacturing

Axially compressed circular cylindrical shells with large diameter-to-thickness ratios are highly susceptible to local buckling, and their load-carrying capacities are known to be very sensitive to initial geometric imperfections. Hence, severe knock-down factors on their theoretical buckling loads are typically prescribed in design specifications, which greatly impair their structural efficiency. With the aim of enhancing load-bearing resistance and reducing sensitivity to imperfections, the shape optimisation and assessment of compressed free-form wavy cylindrical shells, the realisation of which is now viable through additive manufacturing, are the subject of the present study. The adopted optimisation framework employs the Particle Swarm Optimisation (PSO) algorithm, integrating computer-aided geometric design, nonlinear numerical simulations and imperfection sensitivity analyses. The structural performance of the optimised free-form wavy shells is analysed and compared to that of reference circular shells, as well as other types of non-circular shell profiles, including sinusoidally corrugated shells, Aster shells and stringer-stiffened shells. The optimised free-form wavy shell profiles are shown to exhibit increases in ultimate stress of up to 136% compared with the reference circular shell profiles; in general, greater benefits are achieved for more slender cross-sections. In future work, the proposed optimised shells will be manufactured in stainless steel by means of powder bed fusion (PBF), and their structural performance will be further verified through physical experiments.

A Review of T-spline Surfaces

Background: Curved modeling technology originated from the geometric lofting and design of aircrafts, automobiles and ships. The control points of the traditional B-spline mesh should be placed regularly in whole rows and columns. A T-spline surface is a B-spline surface that allows T-junctions. It can overcome the limitations of traditional B-mesh topology and has its own advantages in surface splicing, surface fining, surface simplification, etc. T-spline has wide application prospects in product modeling, art design, animation production, numerical control machining, volume data expression, and other aspects. Objective: The objective of this paper is to summarize the properties, algorithms, and applications of T-splines. It helps scholars in determining the research status of T-splines and in further exploring the theories related to the applications of T-splines. Methods: This paper reviews the theories on T-splines and their applications from four aspects. First, we discuss the development of the concept, properties, and algorithms of T-splines and the Tspline reconstruction. Then, we conducted an isogeometric analysis using T-splines. Next, we demonstrate the applications of T-splines in actual scenarios. Finally, we present a brief summary of the paper and expectations for the future. Results: The paper provides a brief introduction to the relevant papers on T-splines. The research on T-spline technology is currently active, and there are many studies on T-spline theories and applications. Among these, the spline theory on T-mesh has aroused widespread interest in engineering, especially in Computer-Aided Geometric Design (CAGD) and computer graphics. Conclusion: The T-spline surface is the most important new spline surface in the CADG field since the creation of the B-spline surface and non-uniform rational B-spline surface. Although the surface modeling technology based on the T-spline surface is developing rapidly, there are still some problems that need to be further studied.

Frenet Frames of Trigonometric B'ezier Curves

The geometry of curves and surfaces plays a very important role in computer-aided geometric design (CAGD). The goal of this paper is to construct the Frenet frames of trigonometric B\'ezier curves in Euclidean $ 2 $ and $ 3- $space. Especially, the curvatures of these curves are investigated at the beginning and the ending points.

3D Reconstruction & Modeling of the Traditional Greek Trechadiri: “Aghia Varvara”

3D modeling techniques have grown increasingly prevalent in a variety of disciplines, including cultural heritage and ship design. The methodology used in the 3D reconstruction of a traditional Greek boat with the Trechadiri hull type named “Aghia Varvara” is presented in this study. The original boat was built in 1925 and is characterized as a modern cultural heritage monument by the Greek Ministry of Culture. The digital reconstruction of the boat is explained in detail, including 3D laser scanning and computer aided geometric design (CAGD), as well as the description of the 3D printing process. The boat’s 3D digital model has been used for the enrichment of the NAVS Project’s digital library, demonstrating the unique geometrical, typological, and cultural characteristics of Greek traditional shipbuilding, a living craft which listed on Greece’s National Inventory of Intangible Cultural Heritage.

Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials in computational disciplines, we propose a new generalization of Bernstein–Kantorovich operators involving shape parameters λ, α and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program.

On the Kaczmarz methods based on relaxed greedy selection for solving matrix equation [formula omitted]

For solving a consistent system of linear equations, the Kaczmarz method is a popular representative among iterative algorithms due to its simplicity and efficiency. Based on the Petrov–Galerkin conditions and the relaxed greedy selection strategy, we introduce two kinds of Kaczmarz-type methods to solve matrix equation AXB=C, including randomized and deterministic versions and establish the corresponding convergence theories. We prove that our algorithms all have exponential convergence rates. Then we apply our algorithms to real-world application, such as the tensor product surface fitting in computer-aided geometric design (CAGD), it is shown that our algorithms are more effective than some often used conventional matrix equation solvers.

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

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 C1,C2,GC1GC2, 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 C1,GC1 and GC2 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.

Synthesis of Modeling, Visualization, and Programming in GeoGebra as an Effective Approach for Teaching and Learning STEM Topics

GeoGebra is an interactive geometry, algebra, statistics, and calculus application designed for teaching and learning math, science, and engineering. Its dynamic interface allows its users to accurately and interactively visualize their work, models, and results. GeoGebra employs the synthesis of three key features: modeling, visualization, and programming (MVP). Many studies have shown the positive effects of GeoGebra on the efficiency and effectiveness of learning and teaching topics related to science, technology, engineering, and mathematics. In this study, we discuss how GeoGebra provides an environment for learning that is very interactive and collaborative between the learner and the instructor. We also show how integrating GeoGebra into the learning scheme can help improve the skills and knowledge of school and university students in numerous advanced mathematical courses, such as calculus, mathematical statistics, linear algebra, linear programming, computer-aided design, computer-aided geometric design, analytic and projective geometry, and graphical representation. Therefore, this study shows the effectiveness of GeoGebra and its MVP key features in science and engineering, particularly in topics related to mathematics. Each key feature of GeoGebra is thoroughly analyzed, and further analyses, along with how GeoGebra can be helpful in different topics, are discussed.

≈ g(1,2)*-CLOSED AND ≈ g(1,2)*-OPEN MAPS

General topology plays vital role in many fields of applied sciences as well as in all branches of mathematics. In reality it is used in data mining, computational topology for geometric design and molecular design, computer-aided design, computer-aided geometric design, digital topology, information systems, particle physics and quantum physics etc. By researching generalizations of closed sets, some new separation axioms have been founded and they turn out to be useful in the study of digital topology. Therefore, all bi-topological sets and functions defined will have many possibilities of applications in digital topology and computer graphics.