Ternary Schemes Research Articles

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

A Computational Method for Subdivision Depth of Ternary Schemes

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

The inequalities for the analysis of a class of ternary refinement schemes

The ternary refinement schemes are the generalized version of the binary refinement schemes. This class of the schemes produce the smooth curves with the less number of refinement steps as compared to the binary class of schemes. In this paper, we present the inequalities for the analysis of a class of ternary refinement schemes. There are three simple algebraic expressions in each inequality. Further these algebraic expressions contain only the coefficients used in the refinement rules of the ternary schemes.

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.

Can Ternary Computing Improve Information Assurance?

Modern computer microarchitectures build on well-established foundations that have encouraged a pattern of computational homogeneity that many cyberattacks depend on. We suggest that balanced ternary logic can be valuable to Internet of Things (IoT) security, authentication of connected vehicles, as well as hardware and software assurance, and have developed a ternary encryption scheme between a computer and smartcard based on public key exchange through non-secure communication channels to demonstrate the value of balanced ternary systems. The concurrent generation of private keys by the computer and the smartcard uses ternary schemes and cryptographic primitives such as ternary physical unclonable functions. While general purpose ternary computers have not succeeded in general use, heterogeneous computing systems with small ternary computing units dedicated to cryptographic functions have the potential to improve information assurance, and may also be designed to execute binary legacy codes.

Family of a-Ary Univariate Subdivision Schemes Generated by Laurent Polynomial

The generalized symbols of the family of a-ary (a≥2) univariate stationary and nonstationary parametric subdivision schemes have been presented. These schemes are the new version of Lane-Riesenfeld algorithms. Comparison shows that our proposed family has higher continuity and generation degree comparative to the existing subdivision schemes. It is observed that many existing binary and ternary schemes are the special cases of our schemes. The analysis of proposed family of subdivision schemes is also presented in this paper.

A new class of odd-point ternary non-stationary approximating schemes

In this paper, we establish new family of odd-point ternary non-stationary approximating subdivision schemes by introducing the \(sine\) function in the Lagrange identities. It is to be observed that the limiting conic sections, generated by the proposed schemes compare to the existing non-stationary approximating schemes, have less deviation from being an exact conic sections. Moreover, proposed family of 3-point ternary schemes with fewer initial control points produces better limiting conic sections than other existing schemes. Furthermore, the proposed family of schemes is non-stationary counterpart of the stationary scheme of Hassan and Dodgson (Ternary and three-point univariate subdivision schemes. Nashboro Press, Brentwood, 2003), Lian (Appl Appl Math 3:176–187, 2008), Siddiqi and Rehan (Int J Comput Math 87:1709–1715, 2009), Siddiqi and Rehan (Appl Math Comput 216:970–982, 2010), Mustafa et al. (Am J Comput Math 1:111–118, 2011), Aslam et al. (J Appl Math 2011:13, 2011) and Conti and Romani (J Math Anal Appl 407:443–456, 2013).

The Quaternary Interpolating Scheme for Geometric Design

A novel 4-point interpolating subdivision scheme is presented, which generates the family of C2 limiting curves and its limiting function has support on [−7/3,7/3]. It behaves better than classical 4-point binary and ternary schemes with the same approximation order in many aspects that it has smaller support size, higher smoothness, and is computationally more efficient. The proposed nonstationary scheme can reproduce the functions of linear spaces spanned by {1,sin(αx),cos(αx)} for 0<α<π/3. Moreover, some examples are illustrated to show that the proposed scheme can also reproduce asteroids, cardioids, and conic sections as well.

The <i>m</i>-Point Quaternary Approximating Subdivision Schemes

In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.

A Family of Even-Point Ternary Approximating Schemes

We presented a general formula to generate the family of even-point ternary approximating subdivision schemes with a shape parameter for describing curves. Some sufficient conditions for C0 to C7 continuity and approximation order for certain ranges of parameter are discussed. The proposed even-point ternary schemes compare remarkably with existing even-point ternary schemes because they are able to generate limit functions with higher smoothness and approximation order. In addition, we measured curvature and torsion that assist the quality of subdivided curves.

The Odd-Point Ternary Approximating Schemes

We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.

Shape controlled interpolatory ternary subdivision

Ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same (or higher) smoothness but smaller support. In this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements. We show that both these features can be included in a unique interpolating 4-point subdivision method by means of non-stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes. This is realized by exploiting local shape parameters associated with the initial polyline edges.