Research Article Data Visualization Using Rational Trigonometric Spline

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1 Applied Mathematics Volume Article ID 97 pages Research Article Data Visualization Using Rational Trigonometric Spline Uzma Bashir and Jamaludin Md. Ali School of Mathematical Sciences Universiti Sains Malaysia 8 Penang Malaysia Correspondence should be addressed to Uzma Bashir; missheikh9@gmail.com Received March ; Accepted May Academic Editor: Francisco Chiclana Copyright U. Bashir and J. Md. Ali. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive monotone and constrained curve interpolating schemes by using a C piecewise rational cubic trigonometric spline with four shape parameters are developed. Two of these shape parameters are constrained and the other two are set free to preserve the inherited shape features of the data as well as to control the shape of the curve. Numerical examples are given to illustrate the worth of the work.. Introduction Data visualization the technique of using images to represent information has its history in the days back to the second centuryad.butmostofthedevelopmentsaremadeinthe last couple of centuries predominantly during the last years. It has extensively been used in industrial design image processing computer vision computer aided geometric design computer graphics and many more. Shape preserving interpolation is a powerful tool to visualize the data in the form of curves and surfaces. The problem of curve interpolation to the given data has been studied with various requirements. One may be concerned with the smoothness of the interpolating curves the preservation of the underlying shape features of the data the computational complexity or the fulfillment of certain constraints. Shape preserving signifies preserving the three basic and crucial geometrical features such as positivity monotonicity and convexity of the data. Theseshapecharacteristicscanbeeasilyobservedwhendata arises from a physical experiment. In this case it becomes vital that the interpolant produces curves more smooth and represent physical reality as close as possible. For this purpose designers and engineers want such approximation methods that represent such physical situations accurately. At present spline methods have become the main tools forsolvingthemajorityofproblemsinvolvingtheapproximation of functions which also includes interpolation problems. Many spline functions exist that generate smooth and visually pleasant curves. Sarfraz et al. [] studiedshapepreserving curve interpolation for positive monotone convex data and data lying above a line using a piecewise rational cubic function with four shape parameters. Hussain et al. [] visualized scientific data with shape preserving C rational cubic interpolation by developing positive monotone and constrained data preserving schemes. In recent years polynomial splines and NURBS are replaced by trigonometric splines in order to prevail over the difficulties faced in using the former. Polynomial splines are not able to represent circular arcs and conics which are the most basic geometrical entity in almost every modeling system []. Also conics find a widespread application in optical and telecommunication instruments. NURBS offers great flexibility and precision for handling both analytic and freeform shapes but at the expense of intuitiveness and ease of implementation. A number of authors have contributed to trigonometric splines to represent curves and surfaces [ ]. Lately an alternative approach in shape preserving using the trigonometric splines is introduced. Liu et al. [] studied cubic trigonometric polynomial B-spline curves and surfaces with shape parameter. Ibraheem et al. [] introducedac piecewise rational trigonometric cubic function and piecewise rational trigonometric bicubic function with four shape parameters to visualize the positivity of positive curve and

2 Applied Mathematics surface data. Zhu et al. [] constructedshapepreserving quartic trigonometric interpolation spline curves with shape parameters. The authors obtained F continuous shape preserving spline curves for any shape parameters satisfying the shape preserving conditions. In this paper we present a C piecewise rational cubic trigonometric interpolating curve scheme for positive monotone and constrained data. Four positive shape parameters areusedinthedescriptionoftheinterpolatingscheme.two of these parameters are constrained by simple data-dependent conditions to preserve the inherited shape feature of the datawhiletheothertwoarekeptfreetomodifytheshapeof the positive monotone and constrained curve. The scheme is useful for both equally and unequally spaced data. This paper is arranged as follows. In Section ac piecewise rational cubic trigonometric spline with four shape parameters is constructed. Sections and describe positive monotone and constrained curve interpolation schemes respectively to visualize positive monotone and constrained D data. Section concludes the paper with some future work.. C Piecewise Rational Cubic Trigonometric Spline In this section we develop a C piecewise rational cubic trigonometric spline with four parameters. Two of these parametersarefreethatcanbeusedtopreserveandcontroltheshape oftheinterpolatingcurvewhiletheothertwoarerestricted. Data-dependent constraints are developed for these parameters to preserve the shape characteristics of the data under consideration. Suppose that for knot spacing t <t <t < < t n given data points are defined as {(t i f i ): i =... n} where f i are the function values at the knots of the function being interpolated. In each subinterval [t i t i+ ] i =...n a piecewise rational cubic trigonometric spline with four shape parameters is defined as where P (t) P i (t) = j= ξ jb j j= wj i b () j b = ( sin u) b =sin u ( sin u) b =cos u ( cos u) b = ( cos u). The spline in ()isc continuous if the following interpolating conditions are satisfied: P(t i )=f i P(t i+ )=f i+ P (t i )=d i P (t i+ )=d i+ () () where P (t) denotes the derivative with respect to t andd i are derivative values at given knots t i that are used for the shape control and smoothness of curve. These d i are either given or can be computed by some numerical method []. Using conditions () the values of unknowns ξ i i = are ξ =w i f i ξ = w i f i +h i d i w i ξ = w i f i+ h i d i+ w i ξ =w i f i+. Thus after using () ()takestheform where () P (t) P i (t) = p i (u) q i (u) () p (u) =w i f ib +( f i +h i d i w i +( w i f i+ h i d i+ w i )b )b +w i f i+b q (u) =b w i +b +b +b w i u = (/)((t t i )/h i ) h i =t i+ t i andw j i j = are positive shape parameters. It is to mention that if the valuesoftheshapeparametersarechosenontrialbasisthe shape characteristics of the data are not always preserved. Thus there arises a need for some conditions to be imposed on these shape parameters.. Positive Curve Interpolation In this section we utilize C piecewise rational cubic trigonometric spline developed in Section to generate a positivity preserving curve using a positive data set. Theorem. A C piecewise rational cubic trigonometric spline defined in () preserves the positivity of the positive data in each subinterval [t i t i+ ] i =... n iftheshape parameters satisfy the following conditions: w i w i > > max { d ih i w i f i } > max { d i+h i w i f i+ }. Proof. Consider a data set {(t i f i ): i=...n}such that () (7) t i <t i+ i=...n f i >. (8)

3 Applied Mathematics Table : A D positive dataset. i 7 8 t i f i Table : A D positive dataset. i t i.. 7. f i C piecewise rational cubic trigonometric spline given in () preserves positivity through positive data if P i (t) >. P i (t) > if p i (u) q i (u) >. (9) Since q i (u) > thus the positivity of the interpolant depends on p i (u) only. p i (u) > if all the coefficients are positive. It yields > d ih i w i > d i+h i w i. () f i f i+ Thus the sufficient conditions for the interpolant defined in () to preserve the positivity of positive data are that the shape parameters satisfy This proves the desired result. w i w i > > max { d ih i w i f i } > max { d i+h i w i f i+ }. () These conditions on the shape parameters can also be expressed as =η i + max { d ih i w i f i } η i > = ] i + max { d i+h i w i f i+ } ] i >. () The developed scheme is used to demonstrate the positivity preserving of positive data. The curves in Figures and are drawn by using C piecewise rational cubic trigonometric spline for D positive data sets given in Tables and respectively. Random values to the shape parameters are assigned and it is clearly visible that the resulting curves do not preserve the positivity. On the other hand the positivity preserving curves in Figures and are generated for the same data set by using the scheme developed in Section. These figures also delineate the role of free shape parameters in shape control of the curve.. Monotone Curve Interpolation ADdataset{(t i f i ): i =... n} with t i < t i+ i =...n is said to be monotonically increasing (monotonically decreasing) if f i f i+ (f i f i+ ). () wi =. wi =. =. w i = Figure : Nonpositivity preserving rational cubic trigonometric curve. This section discusses a monotonicity preserving curve interpolating scheme with four parameters. For a given set of monotone data points we wish to generate a piecewise cubic trigonometric curve that interpolates the data points and preserves monotonicity as well. We drive data-dependent conditions for two shape parameters while leaving the other two for designer s choice. Theorem. The C piecewise rational cubic trigonometric spline defined in () preserves the monotonicity through monotone data in each subinterval [t i t i+ ] i =...n if the shape parameters satisfy the following conditions: where w i w i > α i < <β i γ i < <δ i α i = max { w i d iw i Δ i } β i =(+ d i Δ i )w i () γ i = max { w i d i+w i Δ i } δ i =(+ d i+ Δ i )w i. ()

4 Applied Mathematics wi =. wi =. wi =. wi =. Figure : A C positivity preserving curve with different values of free parameters. wi = wi =. wi =. wi = Figure : Nonpositive rational cubic trigonometric curve. Proof. Let {(t i f i ): i =... n} be a monotonically increasing data set that is f i f i+ or equivalently Δ i = f i+ f i h i. () The case of monotonically decreasing data set can be dealt in a similar fashion. For monotonicity the necessary conditions on derivatives are d i i=...n. (7) There arise the following two cases for the interpolant () to preserve the monotonicity of monotone data. Case. d i =d i+ =when Δ i =.InthiscaseP i (t) reduces to P i (t) =f i t [t i t i+ ]. (8) Thisprovesthattheinterpolantismonotone. Case.WhenΔ i =thenp i (t) is monotonically increasing if and only if P i (t) > t [t it i+ ]. (9) For t [t i t i+ ] P i (t) is presented in a simpler form as P i (t) = (q i (u)) {B cos u( sin u) +B cos u(+sin u) +B cos u+b cos u sin u+b sin u +B sin u(+cos u) + B 7 sin u( cos u) +B 8 } ()

5 Applied Mathematics wi =. wi =. wi =. wi =. Figure : Positive curve by rational cubic trigonometric spline with different values of free parameters. 8 8 B =(Δ i (w i )+d iw i )w i B =Δ i (w i )(w i w i ) + d iw i (w i w i ) d i+w i (w i ) +(Δ i ( w i ) (d i + d i+ )w i )w i +Δ i w i wi =.7 wi =. wi =. wi =.8 B =(Δ i (w i )+d i+w i )w i B =Δ i ( w i )(w i w i ) Figure : Nonmonotonicity preserving curve. d iw i (w i w i ) d i+w i ( w i ) with B = d i(w i ) B 7 = d i+(w i ) B 8 =Δ i (w i )(w i w i ) B =Δ i ( w i ) (w i w i ) d i+w i ( w i ) d iw i (w i w i ) + d iw i (w i w i ) d i+w i (w i ). ()

6 Applied Mathematics w i =. w i =. w i = w i = Figure : Monotonicity preserving curve with different values of shape parameters Table : A D monotone dataset. i t i. 7 f i Table : A D monotone dataset. i t i 8 9 f i. 8 Also B i i = 8 if >w i w i = =. wi =. wi = > d iw i Δ i () Figure 7: Nonmonotonicity preserving curve. <(+ d i Δ i )w i The denominator of ()is always positive.thus the sufficient conditions for monotonicity preserving curve are since B B 7 >. B k k=...8 () >w i > d i+w i Δ i <(+ d i+ Δ i )w i ()

7 Applied Mathematics w i =. w i = w i = w i = Figure 8: Monotone data visualization with specified values of free parameters. Hence to preserve the monotonicity of monotone data and control the shape of the curve as per desire ()and()can be written as w i w i > 7 α i < <β i () with γ i < <δ i α i = max { w i d iw i Δ i } β i =(+ d i Δ i )w i γ i = max { w i d i+w i Δ i } δ i =(+ d i+ Δ i )w i () as required. To produce a monotone curve using a monotone data the restrictions on the shape parameters can be rearranged as y =.x w i =. =. wi =. wi =. Figure 9: Rational cubic trigonometric curve lying below the given line.. w i w i > α i +k i = =β i l i k i l i > γ i +m i = =δ i n i m i n i >. (7) To implement the developed scheme two monotone data sets are given in Tables and respectively.initiallythe curves are generated by assigning arbitrary values to the four shape parameters and nonmonotonicity preserving curves are shown in Figures and 7respectively.Toremedy this deficiency the scheme developed in Theorem is applied to the same data sets and monotonicity preservation and smoothness of the curves are shown in Figures and 8 respectively.

8 8 Applied Mathematics 7 7 y =.x +. y =.x w i = w i =.8 w i =. w i =. Figure : C rational cubic trigonometric curve lying above the given line with different values of free parameters..8.. Table : D data set lying above the line y =.x +.. i t i.. f i Table : D data set lying above the line y =.x +.. i t i..... f i y =.x +. w i = =..... wi =. wi = Figure : Rational cubic trigonometric curve lying below the given line.. Constrained Curve Interpolation In this section we generalize the curve scheme for positive data developed in Section. It is assumed that the data under consideration lies not only above the line y = but also above any arbitrary line y = mx+c.we wish to drive a scheme for generating a curve which interpolates this data and lies above the line as well. Theorem. The C piecewise rational cubic trigonometric spline defined in () preservestheshapeofdatalyingabove an arbitrary straight line in each subinterval [t i t i+ ] i =... n if the following conditions are satisfied: w i w i > > max { > max { d i h i w i (f i l) } d i+ h i w i (f i+ l) }. (8) Proof. Let {(t i f i ): i =...n} be a set of data points lying above a given straight line y=mx+c;thatis f i >mt i +c. (9)

9 Applied Mathematics y =.x y =.x w i =. w i =. w i =.8 w i =.9 Figure : C piecewise rational cubic trigonometric curve constrained by a given line. The curve will lie above the straight line if the rational cubic trigonometric spline () satisfies the following condition: P (t) >mt+c t [t t n ]. () For each subinterval [t i t i+ ]() can be expressed as P i (t) = p i (u) q i (u) >α i ( u) uβ i () or p i (u) lq i (u) > () where l= α i ( (/)u) (/)uβ i with α i =mt i +cand β i =mt i+ +c.usingthevaluesofp i (u) and q i (u) as defined in () () can be written in a simplified form as Using () we get where j= ξ j b j l j= j= w j i b j >. () b j C j > () C =f i lw i C = w i f i +h i d i w i C = w i f i+ h i d i+ w i C =f i+ lw i. l l () Since b j j = thus()istrueifc j >. As C C >: C >if Also C >if > d ih i w i (f i l). () > d i+h i w i (f i+ l). (7) Thus for a curve constrained by a line the parameters must satisfy w i w i > > max { > max { d i h i w i (f i l) } d i+ h i w i (f i+ l) }. Equation (8) can also be expressed as =μ i + max { = ] i + max { w i w i > d i h i w i (f i l) } μ i > d i+ h i w i (f i+ l) } ] i >. (8) (9)

10 Applied Mathematics The usefulness of the developed scheme is shown by taking data sets lying above a given line. The data set in Table lies above line y =.x +. whereas the data set given in Table lies above the line y =.x +.. Figures 9 and areproducedbytakingthevaluesoftheshape parameters on trial and error basis. These figures depict that thecurvesdonotlieabovetherespectivegivenstraightlines. To remove this drawback curves in Figures and are generated by using the constrained curve scheme developed in the previous theorem. It is clearly shown that the curves not only lie above their same respective lines but also can be made as smooth as required.. Conclusion and Future Plan A C piecewise rational cubic trigonometric spline is discussed in this paper to address the problem of scientific data visualization. Four positive shape parameters are used in the description of positive monotone and constrained curve interpolation schemes. Two of these four shape parameters areconstrainedtopreservetheshapeofdataandtheother two are left free for the designer to alter the shape of curves in order to look like as he wants them to be. The presented scheme works well for both equally and unequally spaced data. It is tested for different data sets to show its usefulness in curve construction. In future this scheme will be extended to rational bicubic surface interpolation scheme to generate positive monotone and constrained surfaces. [8] X. Han Quadratic trigonometric polynomial curves concerning local control Applied Numerical Mathematics vol.no. pp.. [9] X. Han A class of general quartic spline curves with shape parameters Computer Aided Geometric Design vol.8no. pp.. []H.LiuL.LiD.ZhangandH.Wang Cubictrigonometric polynomial B-spline curves and surfaces with shape parameter Information and Computational Sciencevol.9no. pp [] M. Sheng and B. Su A class of Bézier-type curves and surfaces by trigonometric polynomials Computational Information Systemsvol.9no.pp [] F. Ibraheem M. Hussain M. Z. Hussain and A. A. Bhatti Positive data visualization using trigonometric function Journal of Applied Mathematics vol.articleid79pages. []Y.ZhuX.HanandJ.Han QuartictrigonometricBézier curves and shape preserving interpolation curves Computational Information Systems vol.8no.pp.9 9. [] M. Sarfraz S. Butt and M. Z. Hussain Visualization of shaped data by a rational cubic spline interpolation Computers & Graphicsvol.no.pp.8 8. Acknowledgments The authors are grateful to the anonymous referees for their valuable comments which improved this paper significantly. This work is supported by School of Mathematical Sciences Universiti Sains Malaysia. References [] M. Sarfraz M. Z. Hussain and M. Hussain Shape-preserving curve interpolation International Computer Mathematicsvol.89no.pp.. [] M. Z. Hussain M. Sarfraz and M. Hussain Scientific data visualization with shape preserving C rational cubic interpolation European Pure and Applied Mathematicsvol.no. pp. 9. [] G.E.FarinCurves and Surfaces for Computer-Aided Geometric Design: A Practical Code Academic Press Princeton NJ USA 99. [] X. Han Quadratic trigonometric polynomial curves with a shape parameter Computer Aided Geometric Design vol. 9no. 7pp.. [] X. Han Piecewise quadratic trigonometric polynomial curves Mathematics of Computation vol.7no.pp [] X. Han Cubic trigonometric polynomial curves with a shape parameter Computer Aided Geometric Designvol.no.pp. 8. [7] X. Han C quadratic trigonometric polynomial curves with local bias Computational and Applied Mathematics vol. 8 no. pp. 7.

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