The Quadratic Trigonometric Bézier Curve with Single Shape Parameter
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1 J. Basc. Appl. Sc. Res., ( , 01 01, TextRoad Publcaton ISSN Journal of Basc and Appled Scentfc Research The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma Bashr a*, Muhammad Abbas a, Mohd Nan Hj Awang b, Jamaludn Md Al a a School of Mathematcal Scences, Unverst Sans Malaysa, Penang 11800, Malaysa b School of Dstance Educaton, Unverst Sans Malaysa, Penang 11800, Malaysa ABSTRACT In ths work, we have constructed a quadratc trgonometrc Bézer curve wth sngle shape parameter whch s analogous to quadratc Bézer curve. We have adjusted the shape of the curve as desred, by smply alterng the values of shape parameter, wthout changng the control polygon. The quadratc trgonometrc Bézer curve can be made closer to the quadratc Bézer curve or nearer to the control polygon than quadratc Bézer curve due to shape parameter. The representaton of ellpse s more accurate and exact by usng quadratc trgonometrc Bézer curve. k The smoothness of curve s C. KEYWORDS:Trgonometrc polynomals, quadratc trgonometrc bass functons, quadratc trgonometrc Bézer curves, shape parameters. 1. INTRODUCTION Curves and surfaces desgn s an mportant topc of CAGD (Computer Aded Geometrc Desgn and computer graphcs. The sgnfcance of trgonometrc polynomals n dverse areas, namely electroncs or medcne s well known [1]. The parametrc representaton of curves and surfaces specally n polynomal form s most sutable for desgn, as the planer curves cannot deal wth nfnte slopes and are axs dependent too. Lately, The parametrc representaton of curves and surfaces wth shape parameters has ganed much attenton of the desgners and several new trgonometrc splnes have been proposed for geometrc modelng n CAGD. Han [] constructed 1 quadratc trgonometrc polynomal curves wth a shape parameter whch are C contnuous and smlar to quadratc B-splne curves. Wu. X., et al [3] presented quadratc trgonometrc splne curves wth multple shape parameters. Han, X. [4] dscussed pecewse quadratc trgonometrc polynomal curves wth C contnuty. Han, X. [5] studed Cubc trgonometrc polynomal curves wth a shape parameter. These papers [4,5] descrbe the trgonometrc polynomal curves wth global shape parameters. The theory of Bézer curves uphold a key poston n CAGD. These are consdered as deal geometrc standard for the representaton of pecewse polynomal curves. In recent years trgonometrc polynomal curves lke those of Bézer type are consderably n dscusson. Wang, L.,et al [6] dscussed the adjustable quadratc trgonometrc polynomal Bézer curves wth a shape parameter. Han, X.A.,et al [7] ntroduced cubc trgonometrc Bézer curves wth two shape parameters. In the lght of [7], Lu, H., et al [8] dscussed study on a class of cubc trgonometrc Bézer curves wth two shape parameters. We [9] constructed quadratc TC- Bézer curves wth shape parameter n a specal space. In ths paper, we present quadratc trgonometrc Bézer curves wth sngle shape parameter. The curve s constructed based on new quadratc trgonometrc polyonomals. It s analogous to ordnary quadratc Bézer curve. Ths curve takes over the exstng curve n the manner that t exactly expresses some quadratc trgonometrc k curves, arc of a crcle and arc of an ellpse, under suffcent condtons. The proposed curve attans C contnuty. The present work s organsed as follows: In secton, the bass functons of the quadratc trgonometrc Bézer curve wth sngle shape parameter are constructed and the propertes of the bass functons are shown. In secton 3, quadratc trgonometrc Bézer curves and ther propertes are dscussed. The effect on the shape of the curve by alterng the value of shape parameter s gven n secton 4 empowered by the graphcal representaton. In secton 5 the representaton of ellpses s presented. The approxmaton of the quadratc trgonometrc Bézer curve to the ordnary quadratc Bézer curve s gven n secton 6. Fnally, concluson of the work and future road map s gven n secton 7.. QUADRATIC TRIGONOMETRIC BASIS FUNCTIONS In ths secton, defnton and some propertes of quadratc trgonometrc bass functons are gven. Defnton.1: For u [0,1], the quadratc trgonometrc bass functons wth sngle shape parameter m, m [ 1,1] are defned as: *Correspondng Author: Uzma Bashr, School of Mathematcal Scences, Unverst Sans Malaysa, Penang 11800, Malaysa, Emal: msshekh9@gmal.com 541
2 Bashr et al., 01 f0 ( u (1 sn u(1 msn u f1 ( u (1 m(cos u sn u 1 f ( u (1 cos u(1 m cos u For m 1, the bass functons are lnear trgonometrc polynomals. Theorem.1: The bass functons (1 have the followng propertes: (a Nonnegatvty: f ( u 0, 0,1, (b Partton of unty: 0 f ( u 1 (c Monotoncty: For the gven value of the parameter m, f ( u s momotoncally decreasng and f ( u s 0 monotonclly ncreasng. (d Symmetry: f ( u; m f (1 u; m, 0,1, Proof: (a For u [0,1] and m [ 1,1], (1 sn 0 u, (1 m sn u 0, (1 cos 0, u (1 m cos u 0, cos 0 u, sn 0 u and (1 m 0. It s obvous that f ( u 0, 0,1, (b f ( u = (1 sn u(1 msn u + (1 m(cos u sn u 1 + (1 cos u(1 mcos u = 1 0 (c Monotoncty of functons s seen n Fg.1. (d For =, f ( u; m (1 cos u(1 mcos u = (1 sn (1 u(1 msn (1 u = f (1 u; m 0 The quadratc trgonometrc bass functons for m 1 (dashed and for m 1 (sold are shown n Fg.1. (1 Fg.1: The quadratc trgonometrc bass functons 3. QUADRATIC TRIGONOMETRIC BÉZIER CURVE We construct the Quadratc Trgonometrc Bézer (.e.qt-bézer curve wth sngle shape parameter as follows: P 0,1, n, we defne the QT- Bézer curve wth sngle shape Defnton 3.1: Gven the control ponts parameter as: 54
3 J. Basc. Appl. Sc. Res., ( , 01 f ( u f ( u P, u [0,1], m [ 1,1] ( 0 The curve defned by ( possesses some geometrc propertes whch can be obtaned easly from the propertes of the bass functons. Theorem 3.1: The QT- Bézer curve have the followng propertes: (a End pont propertes: f (0 P0, f (1 P f '(0 (1 m( P1 P0, f '(1 (1 m( P P1 f ''(0 m P0 (1 m P1 (1 m P, f ''(1 m P (1 m P1 (1 m P0 f '''(0 (1 m( P1 P0, f '''(1 (1 m( P P1 ( v ( v f (0 8 m P0 (1 m P1 (7m 1 P, f (1 8 m P (1 m P1 (7m 1 P0 Remark (k 1 k (k 1 k ( f (0 ( 1 f (0 and f (1 ( 1 f (1, k 1,, 3,..., n ( Invertng the order of the control ponts, all dervatves of order k, k 1,, 3,..., n concde. (b Symmetry: P0, P1, P and P, P1, P 0 defne the same curve n dfferent parametrzatons, that s: f ( u; m, P0, P1, P f (1 u; m, P, P1, P0, u[0,1], m[ 1,1] (c Geometrc nvarence: The shape of the curve ( s ndependent of the choce of coordnates,.e., t satsfes the followng two equatons: f ( u; m, P q, P q, P q f ( u; m, P, P, P q f ( u; m, P * T, P * T, P * T f ( u; m, P, P, P * T, u[0,1], m[ 1,1] Where q s any arbtrary vector n and T s an arbtrary d d matrx, d (d Convex hull Property: From the non-negatvty and partton of unty of bass functons, t follows that the whole curve s located n the convex hull generated by ts control ponts. Fg. The effect on the shape of the QT- Bézer curve for dfferent values of m 4. SHAPE CONTROL OF THE QT- BÉZIER CURVE The parameter m controls the shape of the curve (. The QT- Bézer curve f(u gets closer to the control polygon as the value of the parameter ncreases gradually n [ 1,1]. In Fg, the curves are generated by settng the values of m as m 0.5 (green dashed dotted, m 0 (black dashed, m 0.5 (pnk dotted, and m 1 ( red sold. The curve goes back to the straght lne when m 1 (blue sold. 5. THE REPRESENTATION OF ELLIPSE Theorem 5.1: Let P0, P1, P be three control ponts on an ellpse wth sem major and mnor axes as a and b respectvely. By the proper selecton of coordnates, ther coordnates can be wrtten n the form 543
4 Bashr et al., 01 a 0 a P0 P1 P 0 b 0 Then for the value of shape parameter m 0 and local doman u [0, 4], the curve ( represents arc of an ellpse wth x( u a (cos u sn u (3 y( u b(cos u sn u 1 Proof: a 0 a If we plug m 0 and P0 P1 P nto (, then the coordnates of QT- Bézer curve are 0 b 0 x( u a (cos u sn u y( u b(cos u sn u 1 Ths gves the ntrnsc equaton x y b ( ( 1 a b It s an equton of the ellpse centered at (0,-b. For u [0, 4], equaton (3 represents the whole ellpse. Fg.3 shows the representaton of ellpse wth QT- Bézer curves Fg 3: The representaton of ellpses wth QT- Bézer curves 6. APPROXIMABILITY Control polygons provde an mportant tool n geometrc modelng. It s an advantage f the curve beng modelled tends to preserve the shape of ts control polygon. The QT- Bézer curve wth sngle shape parameter s analogous to ordnary quadratc Bézer curve wth same control ponts. Theorem 6.1: Suppose P0, P1, P are not collnear; the relatonshps between QT- Bézer curve f(u and the quadratc Bézer curve B( t P (1 t t, t[0,1] wth the same control ponts P 0,1, are as follows: 0 f (0 B(0 f (1 B(1 1 1 f ( P1 ( 1( m( B( P1 (4 544
5 J. Basc. Appl. Sc. Res., ( , 01 Proof: By smple computatons, f (0 P0 B(0, f (1 P B(1 Snce B( t (1 t P0 (1 t tp1 t P So, 1 1 B( P1 ( P0 P1 P 4 and f ( P1 ( 1( m P0 ( 1( m P1 ( 1( m P = 1 ( 1( m ( P 0 P1 P = ( 1( m( B( 1 P1 Corollary 6.1: The QT- Bézer curve s closer to the control polygon than the quadratc Bézer curve f and only f 1 m Corollary 6.: When m, the QT- Bézer curve s close to quadratc Bézer curve,.e. f ( B. Fg. 4 shows the relatonshp between the QT- Bézer curves and quadratc Bézer curves. The QT- Bézer 1 curve(black dotted wth shape parameter m s analogous to ordnary quadratc Bézer curve (pnk sold 7. CONCLUSION Fg 4: The relatonshp between the QT- Bézer curves and ordnary quadratc Bézer curves In ths paper, we have presented the QT- Bézer curve wth sngle shape parameter. All physcal propertes of QT-Bézer curve are smlar to the ordnary quadratc Bézer curve. Due to the shape parameter, It s more useful n font desgnng as compared to ordnary quadratc Bézer curve. We can deal precsely wth crcular arcs wth the help of QT-Bézer curve. The curve exactly represents the arc of an ellpse, the arc of a crcle under certan condtons. The curve acheves C k contnuty. Further more, t s analogous n structure to quadratc Bézer curve. It s not dffcult to adapt a QT-Bézer curve to CAD/CAM system because the quadratc Bézer curve s already n use. In future, we would extend t to cubc Bézer curve and ts surfaces. ACKNOWLEDGEMENTS The authors are hghly oblged to the anonymous referees. for ther valuable comments whch mproved our manuscrpt sgnfcantly. Ths work was supported by FRGS Grant. No.03/PJJAUH/ from the Unverst Sans Malaysa and Government of Malaysa. 545
6 Bashr et al., 01 REFERENCES [1] Hoschek, J., D. Lasser, and L.L. Schumaker,1993, Fundamentals of computer aded geometrc desgn. Vol AK peters Wellesley, MA. [] Han, X., 00, Quadratc trgonometrc polynomal curves wth a shape parameter. Computer Aded Geometrc Desgn, 19(7: [3] Wu, X., X. Han, and S. Luo.,007, Quadratc Trgonometrc Splne Curves wth Multple Shape Parameters. The Proceedngs of 10th IEEE Internatonal Conference on Computer Aded Desgn and Computer Graphcs, Bejng, Oct 15-18: [4] Han, X., 003, Pecewse quadratc trgonometrc polynomal curves. Mathematcs of computaton, 7(43: [5] Han, X., 004, Cubc trgonometrc polynomal curves wth a shape parameter. Computer Aded Geometrc Desgn, 1(6: [6] WANG, L. and X. LIU, 007, Quadratc TC-Bézer curves wth shape parameter. Computer Engneerng and Desgn, [7] Han, X.A., Y.C. Ma, and X.L. Huang, 009, The cubc trgonometrc Bézer curve wth two shape parameters. Appled Mathematcs Letters, (: [8] Lu, H., L. L, and D. Zhang, 011, Study on a Class of TC-Bézer Curve wth Shape Parameters. Journal of Informaton & Computatonal Scence, 8(7: [9] We X. Xu, L. Qang Wang, Xu Mn Lu, 011, Quadratc TC-Bézer Curves wth Shape Parameter, Advanced Materals Research (Volumes :
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