Review of Subdivision Schemes and their Applications

2021 ◽  
Vol 16 ◽  
Author(s):  
Yan Liu ◽  
Huahao Shou ◽  
Kangsong Ji
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Hot Spot ◽  
Geometric Constraints ◽  
Future Application ◽  
Subdivision Surfaces ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Curves And Surfaces ◽  
Subdivision Curves

Background: Subdivision surfaces modeling method and related technology research gradually become a hot spot in the field of computer-aided design(CAD) and computer graphics (CG). In the early stage, research on subdivision curves and surfaces mainly focused on the relationship between the points, thereby failing to satisfy the requirements of all geometric modeling. Considering many geometric constraints is necessary to construct subdivision curves and surfaces for achieving high-quality geometric modeling. Objective: This paper aims to summarize various subdivision schemes of subdivision curves and surfaces, particularly in geometric constraints, such as points and normals. The findings help scholars to grasp the current research status of subdivision curves and surfaces better and to explore their applications in geometric modeling. Methods: This paper reviews the theory and applications of subdivision schemes from four aspects. We first discuss the background and key concept of subdivision schemes. We then summarize the classification of classical subdivision schemes. Next, we show the subdivision surfaces fitting and summarize new subdivision schemes under geometric constraints. Applications of subdivision surfaces are also discussed. Finally, this paper gives a brief summary and future application prospects. Results: Many research papers and patents of subdivision schemes are classified in this review paper. Remarkable developments and improvements have been achieved in analytical computations and practical applications. Conclusion: Our review shows that subdivision curves and surfaces are widely used in geometric modeling. However, some topics need to be further studied. New subdivision schemes need to be presented to meet the requirements of new practical applications.

scholarly journals A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 66 ◽  
Author(s):  
Aamir Shahzad ◽  
Faheem Khan ◽  
Abdul Ghaffar ◽  
Ghulam Mustafa ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Numerical Algorithm ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Curves And Surfaces ◽  
Geometrical Shapes ◽  
Subdivision Depth ◽  
Subdivision Curves

Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.

scholarly journals Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehwish Bari ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
Dumitru Baleanu
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Tension Control ◽  
Free Form ◽  
Subdivision Surfaces ◽  
Smooth Curves ◽  
Practical Applications ◽  
Tension Parameter ◽  
Exact Reproduction ◽  
Free Form Surfaces

AbstractSubdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

scholarly journals Curve and surface construction based on the generalized toric-Bernstein basis functions

2020 ◽  
Vol 18 (1) ◽  
pp. 36-56 ◽  
Author(s):  
Jing-Gai Li ◽  
Chun-Gang Zhu
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Basis Functions ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Parametric Curve ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Bernstein Basis Functions

Abstract The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

Geometric modeling of lattice structures for additive manufacturing

2018 ◽  
Vol 24 (2) ◽  
pp. 351-360 ◽  
Author(s):  
Gianpaolo Savio ◽  
Roberto Meneghello ◽  
Gianmaria Concheri
Keyword(s):  
Additive Manufacturing ◽  
Computer Aided Design ◽  
Geometric Modeling ◽  
Lattice Structure ◽  
Subdivision Surfaces ◽  
Lattice Structures ◽  
Content Type ◽  
Critical Issues ◽  
Surface Subdivision ◽  
Manufacturing Technologies

Purpose This paper aims to propose a consistent approach to geometric modeling of optimized lattice structures for additive manufacturing technologies. Design/methodology/approach The proposed method applies subdivision surfaces schemes to an automatically defined initial mesh model of an arbitrarily complex lattice structure. The approach has been developed for cubic cells. Considering different aspects, five subdivision schemes have been studied: Mid-Edge, an original scheme proposed by the authors, Doo–Sabin, Catmull–Clark and Bi-Quartic. A generalization to other types of cell has also been proposed. Findings The proposed approach allows to obtain consistent and smooth geometric models of optimized lattice structures, overcoming critical issues on complex models highlighted in literature, such as scalability, robustness and automation. Moreover, no sharp edge is obtained, and consequently, stress concentration is reduced, improving static and fatigue resistance of the whole structure. Originality/value An original and robust method for modeling optimized lattice structures was proposed, allowing to obtain mesh models suitable for additive manufacturing technologies. The method opens new perspectives in the development of specific computer-aided design tools for additive manufacturing, based on mesh modeling and surface subdivision. These approaches and slicing tools are suitable for parallel computation, therefore allowing the implementation of algorithms dedicated to graphics cards.

4D printing technology, modern era: A short review

2020 ◽  
pp. 92-111
Author(s):  
Khodadad Mostakim ◽  
Nahid Imtiaz Masuk ◽  
Md. Rakib Hasan ◽  
Md. Shafikul Islam
Keyword(s):  
Smart Materials ◽  
Computer Aided Design ◽  
Short Review ◽  
Stimuli Responsive ◽  
4D Printing ◽  
Space Applications ◽  
Printing Technology ◽  
Practical Applications ◽  
Biomedical Technologies ◽  
Novel Material

The advancement in 3D printing has led to the rapid growth of 4D printing technology. Adding time, as the fourth dimension, this technology ushered the potential of a massive evolution in fields of biomedical technologies, space applications, deployable structures, manufacturing industries, and so forth. This technology performs ingenious design, using smart materials to create advanced forms of the 3-D printed specimen. Improvements in Computer-aided design, additive manufacturing process, and material science engineering have ultimately favored the growth of 4-D printing innovation and revealed an effective method to gather complex 3-D structures. Contrast to all these developments, novel material is still a challenging sector. However, this short review illustrates the basic of 4D printing, summarizes the stimuli responsive materials properties, which have prominent role in the field of 4D technology. In addition, the practical applications are depicted and the potential prospect of this technology is put forward.

scholarly journals Advances in Engineering Graphics: Improvements and New Proposals

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 827
Author(s):  
José Ignacio Rojas-Sola
Keyword(s):  
Computer Animation ◽  
Computer Aided Design ◽  
Geometric Modeling ◽  
Correct Interpretation ◽  
Special Issue ◽  
Engineering Graphics ◽  
Communication Techniques ◽  
Graphic Communication ◽  
Graphic Language ◽  
Aided Design

The study of graphic communication techniques that engineers, architects, and designers use to express ideas and concepts, or the graphic expression applied to the design process, is becoming increasingly important. The correct interpretation of graphic language allows the development of skills in the training of an engineer or architect. For this reason, research on this topic is especially valuable in finding improvements or new proposals that help toward a better understanding of those techniques. This Special Issue shows the reader some examples of different disciplines available, such as engineering graphics, industrial design, geometric modeling, computer-aided design, descriptive geometry, architectural graphics and computer animation.

Boundary Surface Recovery From Skeleton Elements: Part I — Skeleton Curves

1992 ◽  
Author(s):  
Sean M. Gelston ◽  
Debasish Dutta
Keyword(s):  
Inverse Problem ◽  
Computer Aided Design ◽  
Boundary Surface ◽  
Reconstruction Algorithm ◽  
Companion Paper ◽  
Active Area ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Aided Design ◽  
Boundary Surfaces

Abstract Skeleton curves and surfaces have many applications in computer aided design and analysis. Construction of skeletons is an active area of research. We consider the inverse problem that of recovering boundary surfaces from given skeleton elements. The skeleton of any 3D object will, in general, consist of curves and surfaces. Therefore, any boundary reconstruction algorithm must systematically process the surfaces generated by the skeletal curves and the skeletal surfaces. In this paper (Part I) we present algorithms for reconstructing boundary surfaces corresponding to skeletal curves. Implemented examples are also included. In a companion paper (Part II) we consider skeletal elements that are surfaces.

Geometric Modeling of Ceramics Dendrites to Modulate Energy and Material Flows by Using Stereolithography

2014 ◽  
Vol 783-786 ◽  
pp. 2439-2444 ◽  
Author(s):  
Soshu Kirihara
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Lattice Structure ◽  
Photonic Band Gap ◽  
Ultra Violet ◽  
Terahertz Time Domain Spectroscopy ◽  
Cross Sectional ◽  
Air Atmosphere ◽  
Micro Patterning ◽  
Micro Pattern

Through computer aided design, manufacturing and evaluation, various ceramics dendrites with spatially ordered micro cavities were successfully fabricated by utilizing stereolithography. Micrometer order ceramic lattices were propagated spatially in computer graphic space. Ceramics nanoparticles were dispersed in to photo sensitive liquid resins to obtain thixotropic slurries. The paste material was spread on a grass substrate by using a mechanical knife edge, and an ultra violet micro pattern was exposed to create cross sectional solid layer. After the layer stacking process, the obtained composite precursor was dewaxed and sintered in an air atmosphere. By the micro patterning stereolithography, solid electrolyte dendrites of yttria stabilized zirconia with spatially ordered porous structures were fabricated for fuel cell miniaturizations. Gaseous fluid profiles and pressure distributions in the formed ceramic lattices with various porosity percent were visualized and analyzed by a finite element method. Subsequently, alumina micro photonic crystals with a diamond lattice structure were fabricated. Electromagnetic wave properties were measured by using a terahertz time domain spectroscopy. A complete photonic band gap was exhibited, and a localized mode to select the wavelength was obtained by introducing a defect cavity.

scholarly journals A Family of Integer-Point Ternary Parametric Subdivision Schemes

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ghulam Mustafa ◽  
Muhammad Asghar ◽  
Shafqat Ali ◽  
Ayesha Afzal ◽  
Jia-Bao Liu
Keyword(s):  
General Formula ◽  
Integer Point ◽  
Laurent Polynomial ◽  
Polynomial Method ◽  
Subdivision Schemes ◽  
Smooth Curves ◽  
Curves And Surfaces

New subdivision schemes are always required for the generation of smooth curves and surfaces. The purpose of this paper is to present a general formula for family of parametric ternary subdivision schemes based on the Laurent polynomial method. The different complexity subdivision schemes are obtained by substituting the different values of the parameter. The important properties of the proposed family of subdivision schemes are also presented. The continuity of the proposed family is C 2 m . Comparison shows that the proposed family of subdivision schemes has higher degree of polynomial generation, degree of polynomial reproduction, and continuity compared with the exiting subdivision schemes. Maple software is used for mathematical calculations and plotting of graphs.

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