### Computer Graphics and Geometric Modeling Using Beta-splines

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Sign In. Access provided by: anon Sign Out. Rational Beta-splines for representing curves and surfaces Abstract: The rational Beta-spline representation, which offers the features of the rational form as well as those of the Beta-spline, is discussed. The rational form provides a unified representation for conventional free-form curves and surfaces along with conic sections and quadratic surfaces, is invariant under projective transformation, and possesses weights, which can be used to control shape in a manner similar to shape parameters.

Shape parameters are an inherent property of the Beta-spline and provide intuitive and natural control over shape. The Beta-spline is based on geometric continuity, which provides an appropriate measure of smoothness in computer-aided geometric design. The scheme offers a possible and feasible way in which the shape of the objects may be altered by the user. Rational quadratic spline: a Rational quadratic environment. The changes will be local and that the shape will change in a stable manner.

The authors are thinking to extend the scheme for various applications including font de- signing, image outline capture, modeling animation paths, and others. Figure 7.

## A quadratic trigonometric spline for curve modeling

Rational bi-quadratic spline interpolant: a Ra- 7. Acknowledgements tional bi-quadratic 1. Gregory, and P. Yuen, An arbitrary mesh network scheme using rational splines, in: T. Lyche and L.

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Schumaker eds. Gregory, M. Sarfraz, and P. Rational bi-quadratic spline for the design of a , Sarfraz, M. Hussain, and Z. Habib, Local convexity uv-lines and hidden surface removal. Foley and H. Ely, Interpolation with interval and [18] T. Goodman, Properties of Beta-Splines, Journal of point tension controls using cubic weighted Nu-splines, Approximation Theory, 44 2 , Barsky and J. Beatty, Local control of bias and tension , Piegl, and W. Joe, Multiple knot and rational cubic beta-splines, [7] Sarfraz, M. Goodman and K. Kouh, and S. Chau, Computer-aided geometric [22] M.

Sarfraz, Interactive curve modeling with applications design and panel generation for hull forms based on ra- to computer graphics, vision and image processing. Design, , Hussain, M. Irshad, and A. Khalid, Ap- [9] V.

Yahya, A. M Piah and A. Majid, G1 continuity wise Parametric Cubics, Computer Graphics, 17 3 : conics for curve fitting using particle swarm optimization, , Banissi et al. IV, , Graphics and Applications, , Sarfraz, S.

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Raza and M. Baig, Capturing image out- [13] J. Gregory and M. Sarfraz, A rational cubic spline with lines using soft computing approach with conic splines, tension, Computer Aided Geometric Design, , International Conference of Soft Computing and Pattern Recognition, , Ely, Interpolation with interval and [26] P. This is demonstrated in Table 4. A C 2 spline technique QTS is proposed and built with the eagerness of the object modeling using quadratic trigonometric functions.

The proposed scheme is more advantageous over the traditional CPS method. It is smoother, more flexible, faster and more accurate alternate to CPS.

## Computer Graphics and Geometric Modeling Using Beta-Splines

Furthermore, the built curve method is modest overall and is ideal for curve modeling. The authors, as future work, are also looking to expand the idea of the proposed QTS curve models for the designing of surface models. Browse Subject Areas?

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Click through the PLOS taxonomy to find articles in your field. Abstract An imperative curve modeling technique has been established with a view to its applications in various disciplines of science, engineering and design. Data Availability: All relevant data are within the paper.

Funding: The authors received no specific funding for this work. Introduction Designing curves, especially robust curves, which are controllable, well behaved and easily worked out, contributes a special role in computer graphics and geometric modeling. A C 2 QTS can be formed by applying C 2 continuity at the joints of curve segments as follows: 15 From the second order derivative of 1 , we simply achieve the followings: 16 and 17 Let 18 then, the Eqs 16 and 17 , respectively, crop to: 19 and 20 Also 21 Thus, using 19 , 20 and 21 in 14 , a tri-diagonal system of consistency equations is acquired by the followings: 22 The above system can be expressed by the following matrix: 23 It is supposed that end conditions D 0 and D n are given.

Thus, the above conversation can be concise as follows: Theorem 2.

The C 2 QTS exists and has a unique solution. Algorithm design The above discussion is summarized in some steps here. Data of objects The data of various objects drawn in the Figs is given in Table 1. Download: PPT. Time elapsed by the two splines The comparison analysis that the QTS is superior to CPS is also justified by comparing the time of execution of both the splines. Table 3. Advantages of the proposed QTS In this paper, a substantial method has been developed to construct a C 2 QTS and a brief comparison analysis is discussed.

### Table of contents

The advantages of proposed scheme are comprehended as follows: The proposed QTS scheme has decent characteristics of trigonometric splines. The method keeps the suitable geometric properties of splines. It carries out the C 2 smoothness. The proposed QTS produces an alternative to traditional CPS because of having four control points in its piecewise description. Conclusion A C 2 spline technique QTS is proposed and built with the eagerness of the object modeling using quadratic trigonometric functions.

Supporting information. S1 Table. Table of data of objects. References 1. Boehm W.

View Article Google Scholar 2. Barsky B. Thesis, University of Utah, Cline A. View Article Google Scholar 4. Dierckx P. View Article Google Scholar 5.

## Computer Graphics and Geometric Modeling Using Beta-splines : Brian A. Barsky :

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