Computer-aided Geometric Design Research Articles

On G2 approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

Generation of fractal curves using new binary 8-point interpolatory subdivision scheme

The subdivision scheme is a valuable tool for designing shapes and representing geometry in computer-aided geometric design. It has excellent geometric properties, such as fractals and adjustable shape. In this research paper, we explore the generation of fractal curves using a novel binary 8-point interpolatory subdivision scheme with two parameters. We analyse different properties of the proposed scheme, including convergence, special cases, and fractals. Additionally, we demonstrate through various examples the relationship between the shape parameters and the fractal behaviour of the resulting curve. Our research also identifies a specific range of shape parameters that can effectively produce fractal curves. The findings of this study provide a fast and efficient method for generating fractals, as demonstrated by numerous examples. Modelling examples show that the 8-point interpolatory scheme can enhance the efficiency of computer design for complex models.

Generalized Bézier-like model and its applications to curve and surface modeling.

The subject matter of surfaces in computer aided geometric design (CAGD) is the depiction and design of surfaces in the computer graphics arena. Due to their geometric features, modeling of Bézier curves and surfaces with their shape parameters is the most well-liked topic of research in CAGD/computer-aided manufacturing (CAM). The primary challenges in industries such as automotive, shipbuilding, and aerospace are the design of complex surfaces. In order to address this issue, the continuity constraints between surfaces are utilized to generate complex surfaces. The parametric and geometric continuities are the two metrics commonly used for establishing connections among surfaces. This paper proposes continuity constraints between two generalized Bézier-like surfaces (gBS) with different shape parameters to address the issue of modeling and designing surfaces. Initially, the generalized form of C3 and G3 of generalized Bézier-like curves (gBC) are developed. To check the validity of these constraints, some numerical examples are also analyzed with graphical representations. Furthermore, for a continuous connection among these gBS, the necessary and sufficient G1 and G2 continuity constraints are also developed. It is shown through the use of several geometric designs of gBS that the recommended basis can resolve the shape and position adjustment problems associated with Bézier surfaces more effectively than any other basis. As a result, the proposed scheme not only incorporates all of the geometric features of curve design schemes but also improves upon their faults, which are typically encountered in engineering. Mainly, by changing the values of shape parameters, we can alter the shape of the curve by our choice which is not present in the standard Bézier model. This is the main drawback of traditional Bézier model.

A Shape-Preserving Variational Spline Approximation Problem for Hole Filling in Generalized Offset Surfaces

In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces.

Geometric Modelling of a Family of 4-Point Ternary Approximating Subdivision Scheme U_φ with Visual Performance

Making signals better than noise in communication has always been challenging for scientists. Researchers have been working on it in different ways. The computer-aided geometric design is a new research field emerging from the collaboration of computer algorithms and mathematical logic towards curve designing, in which the subdivision schemes used have a key position due to their flexible and smooth behaviour. Using parameters in these schemes allows for increased control over designing. A parameterized framework for generating a wide range of subdivision surfaces with tunable degrees of shape control is presented in the family of schemes. The properties of the proposed family make it suitable for use in isogeometric analysis, computer animation, and geometric modelling. The purpose of this paper is to construct and analyze a family of 4-point ternary subdivision schemes to smooth the curves based on the Laurant polynomial. This family is generated by tuning the weight parameter. The scheme is analysed for its different properties. The scheme has continuity. Visual performance of the subdivision scheme is also provided as an application of this proposed study.

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.

Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications.

In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space-[Formula: see text] with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space-[Formula: see text], and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q.

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.