Approximating Subdivision Schemes Research Articles

A 3-point point quinary approximating subdivision schemes and its application in geometric modeling and computer graphics

This article discusses the significance of a subdivision scheme with shape parameters in geometric modeling and computational geometry. A new recursive method for generating the mask of 3-point quinary approximating subdivision schemes (ASSs) is presented. The proposed subdivision scheme exhibits all the geometric properties and offers better shape control than non-parametric subdivision schemes. The article includes several numerical examples to demonstrate the high practical value of the proposed schemes in geometric modeling and computer graphics.

Convexity and Monotonicity Preservation of Ternary 4-Point Approximating Subdivision Scheme

This work discusses a ternary 4-point approximation subdivision technique with two properties, namely, convexity and monotonicity preservation. The fundamental contribution of this research article is to extract the conditions that assure the suggested subdivision scheme’s convexity and monotonicity. The methodology for extracting these conditions is explained in two theorems. These theorems prove that if the initial data is strictly convex and monotone and the derived conditions are satisfied, then the limiting curve generated by the proposed subdivision scheme will also be convex and monotone. To show the graphical simulations of results, 2D graphs are plotted. Curvature plots are also drawn to fully comprehend the derived conditions. The entire discourse is backed up by convincing examples.

A UNIQUE COMBINATION OF MASK IN BINARY FOUR-POINT SUBDIVISION SCHEME

A unique binary four-point approximating subdivision scheme has been developed in which one part of binary formula have stationary mask and other part have the non-stationary mask. The resulting curves have the smoothness of C3 continuous for the wider range of shape control parameter. The role of the parameter has been depicted using the square form of discrete control points.

Variations in 2D and 3D Models by a New Family of Subdivision Schemes and Algorithms for its Analysis

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision schemes provide the flexibility in designing the limit curves and surfaces. Therefore, the limit curves generated by the proposed subdivision schemes variate in between or around the approximating and interpolatory curves. We also design few analytical algorithms to study the properties of the proposed schemes theoretically. The efficiency of these algorithms is analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed.

Subdivision Collocation Method for One-Dimensional Bratu’s Problem

The purpose of this article is to employ the subdivision collocation method to resolve Bratu’s boundary value problem by using approximating subdivision scheme. The main purpose of this researcher is to explore the application of subdivision schemes in the field of physical sciences. Our approach converts the problem into a set of algebraic equations. Numerical approximations of the solution of the problem and absolute errors are compared with existing methods. The comparison shows that the proposed method gives a more accurate solution than the existing methods.

A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme

In this article, we present a new subdivision scheme by using an interpolatory subdivision scheme and an approximating subdivision scheme. The construction of the subdivision scheme is based on translation of points of the 4-point interpolatory subdivision scheme to the new position according to three displacement vectors containing two shape parameters. We first study the characteristics of the new subdivision scheme analytically and then present numerical experiments to justify these analytical characteristics geometrically. We also extend the new derived scheme into its bivariate/tensor product version. This bivariate scheme is applicable on quadrilateral meshes to produce smooth limiting surfaces up to C 3 continuity.

Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.

Shape Preserving Properties with Constraints on the Tension Parameter of Binary Three-Point Approximating Subdivision Scheme

The paper analyzes conditions for preserving the shape properties from the initial data to the limit curves of the binary three-point approximating subdivision scheme. We provide suitable conditions on the initial data utilizing the tension parameter [Formula: see text], thus the scheme can maintain three important shape properties, namely positivity, monotonicity and convexity in the limit curves. The use of derived conditions is illustrated in few examples, which offers more flexibility in the generation of smooth limit curves endowed with shape preserving properties.

A shape-preserving variant of Lane-Riesenfeld algorithm

<abstract> <p>This paper introduces a family of shape-preserving binary approximating subdivision schemes by applying a shape-preserving variant on the Lane-Riesenfeld algorithm. Using the symbols of subdivision schemes, we determine convergence and smoothness, Hölder continuity, and support size of the limit curves. Furthermore, these schemes produce monotonic and convex curves under the certain conditions imposed on the initial data.</p> </abstract>

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P4, P5, P6, and P7, have C7, C9, C11 and C13 continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

A Family of High Continuity Subdivision Schemes Based on Probability Distribution

Subdivision schemes are famous for the generation of smooth curves and surfaces in CAGD (Computer Aided Geometric Design). The continuity is an important property of subdivision schemes. Subdivision schemes having high continuity are always required for geometric modeling. Probability distribution is the branch of statistics which is used to find the probability of an event. We use probability distribution in the field of subdivision schemes. In this paper, a simplest way is introduced to increase the continuity of subdivision schemes. A family of binary approximating subdivision schemes with probability parameter p is constructed by using binomial probability generating function. We have derived some family members and analyzed the important properties such as continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils. It is observed that, when the probability parameter p = 1/2, the family of subdivision schemes have maximum continuity, generation degree and Holder regularity. Comparison shows that our proposed family has high continuity as compare to the existing subdivision schemes. The proposed family also preserves the shape preserving property such as convexity preservation. Subdivision schemes give negatively skewed, normal and positively skewed behavior on convex data due to the probability parameter. Visual performances of the family are also presented.

Numerical Solution of 2-point Boundary Value Problem by Subdivision Scheme

A numerical approximating collocation algorithm is formulated that is based on binary 6-point approximating subdivision scheme to generate the curves. It is examined that the scheme is generating more smooth continuous solutions of the problems. Numerical example is given to illustrate the algorithm with its graphically representation.

Univariate approximating schemes and their non-tensor product generalization

AbstractThis article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

A combined approximating and interpolating ternary 4-point subdivision scheme

In this paper, a new combined approximating and interpolating ternary 4-point subdivision scheme with multiple parameters is proposed. A set of nice properties, such as support, continuity and polynomial generation, are briefly discussed. The new combined scheme not only contains a lot of classical ternary schemes as special cases, but also generates brand-new ternary schemes. Compared to other approximating subdivision schemes, limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better. Examples are given to show the effectiveness of the scheme. Furthermore, fractal property is analyzed and fractal curves are also given.

Anisotropic bivariate subdivision with applications to multigrid

In this paper, motivated by applications to multigrid, we present families of anisotropic, bivariate, interpolating and approximating subdivision schemes. We study the minimality and polynomial generation/reproduction properties of both families and Hölder regularity of their prominent representatives. From the symbols of the proposed subdivision schemes, we define bivariate grid transfer operators for anisotropic multigrid methods. We link the generation/reproduction properties of subdivision to the convergence and optimality of the corresponding multigrid methods. We illustrate the performance of our subdivision based grid transfer operators on examples of anisotropic Laplacian and biharmonic problems.

Curve Variations in Non-Stationary Three-Point Subdivision Schemes

Subdivision schemes are acknowledged as an important tool in computer aided geometric design. The new binary non-stationary three-point approximating subdivision schemes have been proposed that generate wide variations of C 1 and C 2 continuous curves using shape control parameter &xi 0 . The proposed schemes are the counterpart of stationary schemes introduced by Hormann and Sabin (2008) and Siddiqi and Ahmad (2007). Curve variations using the shape control parameter &xi 0 have been demonstrated by the several examples.

The (2n+1)^2-Point Scheme Based on Bivariate Quartic Polynomial

We are going to implement least squares approach to fit the bivariate quartic polynomial to (2n+1)2- perceptions/data, where n>2. By taking different values of n, (2n+1)2-point approximating subdivision schemes are built. The proposed scheme can be applied for illustrate individual items as a part of 3D (Three Dimensional) space. The proposed scheme is based on fitting the local least squares bivariate quartic polynomial of degree four to the (2n+1)2-observations. The influence of the proposed scheme is shown by 2D example and its working is presented with the help of different quadrilateral meshes. Subdivision and topological rules are also explained with graphical and mathematical representation. Applications and visual exhibitions of the plan have additionally been displayed to show the implementation of the plan.

Fractal Generation by 7-Point Binary Approximating Subdivision Scheme

Fractal Generation by 7-Point Binary Approximating Subdivision Scheme

Generalization of Binary Tensor Product Schemes Depends upon Four Parameters

This article deals with general formulae of parametric and non parametric bivariate subdivision scheme with four parameters. By assigning specific values to those parameters we get some special cases of existing tensor product schemes as well as a new proposed scheme. The behavior of schemes produced by the general formulae is interpolating, approximating and relaxed. Approximating bivariate subdivision schemes produce some other surfaces as compared to interpolating bivariate subdivision schemes. Polynomial reproduction and polynomial generation are desirable properties of subdivision schemes. Capability of polynomial reproduction and polynomial generation is strongly connected with smoothness, sum rules, convergence and approximation order. We also calculate the polynomial generation and polynomial reproduction of 9-point bivariate approximating subdivision scheme. Comparison of polynomial reproduction, polynomial generation and continuity of existing and proposed schemes has also been established. Some numerical examples are also presented to show the behavior of bivariate schemes.

A family of univariate ternary subdivision schemes with 5 parameters

In this paper, we propose an efficient strategy for the construction of a family of ternary subdivision schemes with five parameters. These five parameters allow us to increase the choices when drawing and designing different models from the same initial control points. Moreover, by choosing appropriate values of parameters, we can get many existing ternary approximating subdivision schemes. Some of the existing ternary interpolatory and many new ternary approximating/interpolatory subdivision schemes with arbitrary order of regularities can also be obtained.