Designing by a Quadratic Trigonometric Spline with Point and Interval Shape Control

Transcription

1 Int'l Conf. Scientific Computing CSC' Designing by a Quadratic Trigonometric Spline with Point and Interval Shape Control Shamaila Samreen Department of Mathematics University of Engineering & Technology, Lahore, Pakistan shamailasamreen16@gmail.com Muhammad Sarfraz Dept. of Information Science Kuwait University Safat 13060, Kuwait prof.m.sarfraz@gmail.com Malik Zawwar Hussain Department of Mathematics University of The Punjab Lahore, Pakistan malikzawwar.math@pu.edu.pk Abstract A proficient technique is adopted to construct a spline with quadratic trigonometric functions. The methods is not only endowed with the ability of shape model, also it is accountable for appropriate estimation of the control points. The curve models put up through the developed method; possess the most suitable geometric properties like partition of unity, convex hull, variation diminishing and affine invariance. The graphical depiction of the scheme is ready to lend a hand for various shape influences, like global tension, interval tension or point tension. The developed spline technique is geometrically smooth. Keywords Computer Graphics; Bézier form; trigonometric functions; spline; shape parameters I. INTRODUCTION Designing of curves, specifically vigorous curves, which are straightforwardly puzzle out and manageable shows an exceptional play in CAD/CAM. Numerous applications of these curves in font design, designing objects, fingers print recognition and medical imaging are the enthusiasms in the direction of curve designing. A reasonable amount of work, in the current literature, has been done in the direction of curve designing. For conciseness, the readers are suggested to [1-29]. To be a part of a constructive way out of many challenges at a single platform, it is desired to have a vigorous scheme. Deliberation of a stable trigonometric spline composed with a support of certain added shape parameters and accompanied by piecewise depictions may be a best selection for this paper. It is advisable to put forward a method for a set of trigonometric-splines using the added shape parameters. Foley [11], established a weighted B-spline for cubic function. Nu splines [4, 10] were discovered by cubic spline scheme. In 2004, Sarfraz [21], built weighted Nu splines which was one step ahead to the weighted splines [11] and Nu spline [4]. In 2016, Sarfraz et al. developed the weighted quadratic trigonometric splines to model the objects. Later in 2017 Sarfraz et al. also fabricated the nu spline for trigonometric function having well meticulous shape effects of parameters. An imperious technique has been developed in the paper to build a weighted nu-spline with a quadratic trigonometric function which was richen with the well-meticulous shape effects of the parameters. The developed scheme has subsequent attributes and advantages: It retains appropriate aspects of trigonometric splines. The scheme is talented with fascinated shape modeling features. It sustains the finest appropriate geometric properties of splines. It accomplishes the smoothness geometrically. As a special consideration the scheme recuperates the trigonometric spline. It has two families of shape parameters, which play a part for numerous shape influences, like global tension, interval tension or point tension. For the different choice of parameters, the proposed scheme recovers weighted spline [11] and Nu-spline [4]. In this paper a quadratic trigonometric weighted nu spline (QTWNS) in its interpolation form is presented by inserting weighted nu continuity constraints into the quadratic trigonometric spline. The illustration of the proposed spline is taken into consideration. Furthermore, a proof of affine invariance property for proposed spline curve is produced from Bernstein Bézier form. The graphical depiction of proposed spline subsidizes various shape parameters which can be suitable and specifically supportive for different shape influences, like global tension, interval tension and point tension. The paper has been organized as follows. Section II proposes, describes and analyses the quadratic trigonometric spline in its interpolation form. Section III describes and discusses various geometric properties of the proposed spline.

2 196 Int'l Conf. Scientific Computing CSC'18 The illustration of the offered spline has been depicted in Section IV. Whereas Section V concludes the paper. II. A QUADRATIC TRIGONOMETRIC FORM Let et be the given data points at the knot vectors where. Also let for. The quadratic trigonometric is described for by (1) where,. Rewrite equation (1) as: where, (2) are trigonometric functions with. (3) Furthermore, these functions are Bernstein Bézier weight functions. Also, (4) (5) The function in (1) accomplishes the following properties of interpolation; (6) Thus the interpolation of piecewise function in (1) acts like Hermite interpolation. Hence, by using the subsequent constraints, spline characteristics are incorporated with quadratic trigonometric spline: (7) In this way geometric continuity of 2 nd order at different joins of the curve pieces is achieved to attain a smooth spline. The matrix form of the continuity constraints can be expressed as: (8) To compute the parameters, it is prerequisite to determine the following derivatives, which are given by Since, and (9) (10) (11) the above equations (10) and (11) yield, Also, (12) (13) (14) (15) By using the value of and from (4) and (5), (7) and (8) together with (13), (14) and (15) leads to a tri-diagonal system of consistency equations: (16) which is diagonally dominant for suitable end. It also has unique solutions for To determine the solution of the tridiagonal system for, it is efficient to obtain the LUdecomposition. Thus we conclude the following: Theorem: For the system of equations (16), the quadratic trigonometric spline (QTS) accomplishes a unique solution. III. PROPERTIES The QTS curve satisfies important geometric and shape properties. A. Geometric Properties The QTS curve satisfies the subsequent geometric properties: Proposition 1: The quadratic trigonometric basis functions, have the following properties:

3 Int'l Conf. Scientific Computing CSC' Partition of unity: (17) Positivity: (18) End point interpolation: and (19) Proof: The above properties result immediately from the formation of the basis function. Proposition 2 (Variation Diminishing Property): Consider a QTS curve for with control points. Then any plane of dimension N-1 will intersect the curve no more time it intersects the control polygon. A. Weighted Nu-Spline Let us have the shape parameters: and (20) Now, by applying the subsequent constraints, weighted nu spline characteristics are incorporated with QTS:, In this way the above geometric continuity of 1 st order at different joins of the curve parts is achieved to attain a weighted nu spline curve. The matrix form of the weighted nu-continuity constraints can be presented as: which is diagonally dominant for suitable end conditions. B. Shape Properties The proposed QTWNS is beneficial for numerous applications wherever a point tension and an interval property is required. By changing shape parameters, not only the desired results can be acquired in a specific portion but also whole shape of the object can be change using the QTWNS curve scheme according to one s desire. Figure 1. Variation diminishing property. Proposition 3 (Affine Invariance Property): Let for be QTS curve with control points It fulfills the affine invariance property. Proof: The proof is left as an exercise for the reader. 1. Point Tension Property If for a fixed, the equation (16) consequences as: (21) Thus a corner at will be appeared. 2. Interval Tension Property I In the similar way, if and, the equation (16) consequences as: (22) (23) Thus the curve approaching to the control polygon in the k th interval 3. Global Tension Property Similarly, if Figure 2. Affine invariance property with scaling, rotation and translation, with scaling, shearing and translation. then (24)

4 198 Int'l Conf. Scientific Computing CSC'18 Hence the curve is tightened globally in the interval 4. Interval Tension Property II One can observe that by increasing the value of, the kth segment of the curve is tightened in the k th interval This interval tension is different than the one explained in Interval Tension Property I. It keeps the smoothness of the curve without affecting the corners. Remark (Special Cases): By the uniqueness of shape parameters, elaborated in (17), the proposed spline is one step ahead to the following splines: 1. For and, the proposed spline covers a Quadratic trigonometric spline. 2. For, it is a quadratic trigonometric weighted spline. 3. If,and proposed spline leads to a quadratic trigonometric Nu-spline. IV. DEMONSTRATION The set of data points of different existing objects are interpolated by QTWNS curve. The local and global tension for QTWNS is depicted with the examples. In the graphical depiction, the default value of shape parameter, should be in use, if not then it is described. All over the illustration, the part of the Figures in this section, will be QTWNS interpolation curve with periodic end conditions. Figure 4. (b-d) The QTWNS interpolation with global tension, using and respectively, at all points of the vase. Figure 3. (b-d) The QTWNS interpolation with interval tension, using and respectively, at base intervals of the vase. Figure 5. (b-d) The QTWNS interpolation curve with point tension, using and respectively, at two opposite points of the vase.

5 Int'l Conf. Scientific Computing CSC' Figure 6. (b-d) The QTWNS interpolation curve with interval tension, using and respectively, at four points of neck of the vase. (c) Figure 7. (b-d) The QTWNS interpolation curve with interval tension, using respectively, at the base of the vase. Figures 3, (c) and (d) present interval tensions I at two opposite points of the base of vase with the rising values of as and respectively. In Figures 4(b-d), global tension (d) is provided, by varying as and, respectively, at all points of the vase. Figures 5(b-d) magnificently demonstrate the point tension property in the two opposite points of the vase by varying as and, respectively. In Figures 6, 6(c) and 6(d), interval tension I appears in the neck of the vase with the rising values of as and respectively, in the four points of neck of vase. In Figure 7, 7(c) and 7(d) interval tension II is shown by varying respectively. The difference of interval tension I and interval tension II are very obvious from Figures 3 and 7. V. CONCLUSION A QTWNS curve method has been assembled with the enthusiasm of curve designing towards interpolation of curves. This scheme not only has the proficiency of curve designing, but also it crops up with a suitable estimation to the control points. The proposed method has been explained with the view of its applications in CAGD, CAD/CAM, computer graphics and geometric modeling. A spline method has been developed, which recovers many existing spline methods. The proposed scheme is adequate and mostly supportive for various shape influences, like global tension, interval tension or point tension. In addition shape impacts of parameters are well meticulous for designing purposes. ACKNOWLEDGMENT The authors are grateful to anonymous referees for the helpful comments towards the improvement of the paper. This work has been supported by Kuwait University, Kuwait. REFERENCES [1] I. J. Schoenburg, Contributions to the problem of approximation of equidistant data by analytic functions, Applied Mathematics, 4 (1946) [2] G. M. Nielson, Some piecewise polynomial alternatives to S- splines under tension, Computer Aided Geometric Design, (1974) [3] A. Cline, Curve fitting in one and two dimensions using splines under tension, Communications of the ACM, 17 (1974) [4] J. Lewis, B-spline bases for splines under tension, Nu splines, and fractional order splines, Presented at the SIAM-SIGNUM- Meeting, San Francisco, USA, (1975). [5] B. Barsky, The Beta-spline: A local representation based on shape parameters and fundamental geometric measure, Ph.D. Thesis, University of Utah, (1981). [6] B. Barsky and J. Beatty, Local control of bias and tension in Beta spline, Computer and Graphics, 17(3) (1983) [7] R. Bartels and J. Beatty, Beta-splines with a difference, Technical Report CS-83-40, Computer Science Department, University of Waterloo, Waterloo, Canada, (1984). [8] W. Boehm, Curvature continuous curves and surfaces, Computer Aided Geometric Design, 2(2) (1985) [9] T. N. T. Goodman and K. Unsworth, Generation of Beta spline curves using a recursive relation, Fundamental Algorithms for Computer Graphics, (1985) [10] G. M. Nielson, Rectangular -splines, IEEE Computer Graphics Application, 6 (1986)

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