Cubic Trigonometric Rational Wat Bezier Curves

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1 Cubc Trgonometrc Ratonal Wat Bezer Curves Urvash Mshra Department of Mathematcs Mata Gujr Mahla Mahavdyalaya Jabalpur Madhya Pradesh Inda Abstract- A new knd of Ratonal cubc Bézer bass functon by the blendng of algebrac polynomals trgonometrc polynomals usng weght method s presented named WAT Ratonal Bézer bass are used to developed ratonal cubc Bezer curve. Here weght coeffcents are also shape parameters whch are called weght parameters. The nterval [0 1] of weght parameter values can be extended to [ -.] the correspondng WAT Bézer curves surfaces are defned by the ntroduced new ratonal bass functons. The WAT Bézer curves nhert most of propertes smlar to those of Cubc Bézer curves can be adjusted easly by usng the shape parameter λ. Wth the shape parameter chosen properly the defned curves can express exactly any plane curves or space curves defned by parametrc equaton based on{1 snt cost sntt cost} crcular helx wth hgh degree of accuracy wthout usng ratonal form. Examples are gven to llustrate that the curves surfaces can be used as an effcent new model for geometrc desgn n the felds of CAGD. Ths WAT Bézer curves much closer to the control pont compare to the other curves. The geometrc effect of the of ths weght parameter s dscussed. Index Terms- Bézer curves surfaces trgonometrc polynomal shape parameter G C contnuty. 1. INTRODUCTION Computer aded geometrc desgn (CAGD) studes the constructon manpulaton of curves surfaces usng polynomal ratonal pecewse polynomal or pecewse ratonal methods. Among many generalzatons of polynomal splnes the trgonometrc splnes are of partcular theoretcal nterest practcal mportance. In recent years trgonometrc splnes wth shape parameters have ganed wde spread applcaton n partcular n curve desgn. Bézer form of parametrc curve s frequently used n CAD CAGD applcatons lke data fttng font desgnng because t has a concse geometrcally sgnfcant presentaton. Smooth curve representaton of scentfc data s also of great nterest n the feld of data vsualzaton. Key dea of data vsualzaton s the graphcal representaton of nformaton n a clear effectve manner. In the recent past a number of authors references have contrbuted to the shape-preservng nterpolaton. In [1-6] dfferent polynomal methods whch are used to generate the shape-preservng nterpolant have been consdered. In ths paper we present a class of new dfferent trgonometrc polynomal ratonal bass functons wth a parameter based on the space Ω=span {1 snt cost sntt cost} the correspondng curves tensor product surfaces named WAT Ratonal trgonometrc Bézer curves surfaces are constructed based on the ntroduced new dea of ratonal bass functons. The WAT Ratonal Cubc trgonometrc Bézer curves not only nhert most of the smlar propertes to Cubcc Bézer curves but also can express any plane curves or space curves defned by parametrc equaton based on {1 snt cost sntt cost} ncludng some quadratc curves such as the crcular arcs parabolas cardod exactly crcular helx wth hgh degree of accuracy under the approprate condtons.. In ths paper we present a WAT Ratonal Cubc Trgonometrc Bézer curves wth weght parameter based on the blendng space span. Also the change range of the curves s wder than that of C-Bézer curves. The paths of the gven curves are lne segments. Some transcendental curves can be represented by the WAT wth the shape parameters control ponts chosen properly. The rest of ths paper s organzed as follows. Secton defnes the WAT-Bezer ratonal bass Functons the correspondng curves surfaces thers propertes are dscussed. In secton we dscussed the shape control of the WAT Ratonal Cubc Trgonometrc Bézer curves. In secton 4 Approxmaton of WAT Ratonal Cubc Trgonometrc Bézer curves to the ordnary WAT Cubc Trgonometrc Bézer curve ordnary Cubc Bézer curves are presented. In secton 5 we dscussed contnuty condtons of IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 691

2 WAT-Ratonal Bezer curves. In secton 6 we show the representatons of some curves. Besdes some examples of shape modelng by usng the WAT- Bezer surfaces are presented also. The conclusons are gven n secton 7.. WAT- RATIONAL BASIS FUNCTIONS WAT- BEZIER CURVES AND SURFACES.1 The Constructon of the WAT-Bézer Ratonal Bass Functons The WAT cubc Bezer bass functon defned by Xe et. al. [ ] are as follows: Defnton.1.1: For 0 λ 1 the followng four functons of t [0 1] are defned as WAT cubc Bezer bass functons ( 1 t) snt b0( t ) ( 1 t) ( 1 ) 1 1 snt b1( t ) ( 1 t) t ( 1 ) t cos t 1 1 snt b ( t ) ( 1 t)t ( 1 ) t cos t t snt b 4( t ) t ( 1 ) (1) Defnton.1.: For t [0 1] 0 λ 1 we defne a WAT ratonal cubc Bezer bass functons wth a shape parameter λ: ( 1 t) snt w0 ( 1 t) ( 1 ) b 0( t ) wct snt w1 ( 1 t) t ( 1 ) t cos t b 1( t ) wct snt w ( 1 t)t ( 1 ) t cos t b ( t ) wct 0 t snt w t ( 1) b( t ) wc t 0 () b(t ) Where (=0 1 ) are the WAT cubc Bezer bass functons defned n eq. (1). (a) (b) Fgure 1. (a) For λ = 0.5 (b) For λ = The Propertes of the WAT Ratonal Cubc Bass Functons Theorem 1: The bass functons (.1) have the followng propertes: Straght calculaton testfes that these WAT-Bézer bases have the propertes smlar to the cubc Bernsten bass as follows. 1) Propertes at the endponts: ( j) b 0(0 ) 1 b (0 ) 0 ( j) b (1 ) 1 b (1 ) 0 Where j = = 1 (0) b (t ) b( t ) ) Symmetry: b (t ) b (1 t ) 1 b (t ) b ( 1 t ) 0 IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 69

3 ) Partton of unty: b (t ) 1 0 4) Nonnegatvty: b (t ) 0; 01. Accordng to the method of extendng defnton nterval of C-curves n Ref. the nterval [0 1] of weght parameter values can be extended to where =... WAT- RATIONAL BÉZIER CURVES..1 The Constructon of the WAT- Ratonal Bézer Curves Defnton..1 Gven ponts P k ( k = 0 1 ) n R or R then R(t ) P wb (t ) 0 wb (t ) 0 ;t [01] for = 0 1. [-.] () R( t) Ths s called WAT- Ratonal Bezer curve b (t ) 0; 01. are the WAT-Bezer bass... The Propertes of the WAT- Ratonal Bezer Curve From the defnton of the bass functon some propertes of the WAT- Ratonal Bezer curve can be obtaned as follows: Theorem : The WAT- Ratonal Bezer curve curves (..1) have the followng propertes: Termnal Propertes R(0 ) P 0 R(1 ) P4 ' ( )w R (0 ) 1 (P1 P 0) w0 ' ( )w R (1 ) (P P ) w () Symmetry: Assume we keep the locaton of control ponts P ( = 014) fxed nvert ther orders then the obtaned curve concdes wth the former one wth opposte drectons. In fact from the symmetry of WAT- Bezer base functons we have R(1- t λ P P P 1 P 0 ) = R( t λ P 0 P 1 P P ) ; t [01] λ [-.] Geometrc Invarance: The shape of a WAT- Ratonal Bezer curve s ndependent of the choce of coordnates.e. (..1) satsfes the followng two equatons: R(t; λ; P 0 + q P 1 +q P +q P +q ) = R(1- t; λ; P P P 1 P 0 ) + q ; R(t; λ; P 0 *T P 1 *T P *T P *T ) = R(1- t; λ; P P P 1 P 0 ) * T ; Where q s arbtrary vector n R or R T s an arbtrary d * d matrx d = or. Convex Hull Property: The entre WAT- Ratonal Bezer curve segment les nsde ts control polygon spanned by P 0 P 1 P P.. WAT Ratonal Bézer Surfaces (a) (b) Fgure.(a) For λ = 0.5 (b) For λ = 0.75 P Defnton. Gven the control mesh [ rs ] ( r = + ; s = j j + )( = 0 1 n 1; j = 01 m 1) Tensor product WAT- Ratonal Bézer surfaces can be defned as wb(u λ ) b (v λ )P (u v) 1 r= s= j j r= s=j 1 rs wb(u λ )b (v λ ) R (u v) = ;(u v) Where 1 base functon. (4) b (uλ ) b a (v λ ) are WAT-Bezer IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 69

4 .SHAPE CONTROL OF WAT- RATIONAL BEZIER CURVE Due to the nterval [0 1] of weght parameter values can be extended to[.] the change range of the WAT- Ratonal Bézer curve s wder than that of C- Bézer. From the Fgure t can be seen that when the control polygon s fxed by adjustng the weght parameter from - to. the WAT- Ratonal Bézer curves can cross the cubc Bézer curves reach the both sdes of cubc Bézer curves n other words the WAT- Ratonal Bézer curves can range from below the C-Bézer curve to above the cubc Bézer curve. The weght parameters have the property of geometry. The larger the shape parameter s the more approach the curves to the control polygon s. Also these WAT- Ratonal Bézer curves we defned nclude C-Bézer curve ( α = π ) as specal cases. So that WAT- Ratonal Bezer Curve have more advantages n shape adjustng than that C-Bezer curves do. Fgure. Shows the effect on the shape of WAT-Ratonal Bezer Curve wth alterng the values of λ = - n green λ = 0 n red λ = 1 n blue λ =. n yellow. Fgure. The effect on the shape of WAT-Ratonal Bezer Curve wth alterng the values of λ = - n green 0 n blue 1 n red. 4. APPROXIMABILITY Control polygon provdes an mportant tool n geometrc modelng. It s an advantage f the curve beng modeled tends to preserve the shape of ts control polygon. Now we show the relatons of the cubc trgonometrc ratonal B-splne curve cubc Bézer curves correspondng to ther control polygon. Theorem : Suppose P 0 P 1 P P are not collnear; the relatonshp between cubc trgonometrc ratonal B-splne curve R(t ) (..1) the Ratonal cubc Bézer curve t [0 1] wth the same control ponts P (=01) are gven by B(t) Where P wb (t) 0 wb (t) 0 -j j B (t) = P ( j)(1- t) t j=0 polynomal the scalar functon. w are (5) Bernsten are the weght 1 1 R( ) - P = a B( ) - P ; (6) a 14 8(1 )( ) Where b 14 8(1 )( ) P0 P P Proof: we assume that w0 w 1 w w 1 then the ordnary Ratonal cubc Bezer curve (5) takes the form (1 t) P0 6(1 t) tp1 6(1 t)t P t P B(t) (1 t) 6(1 t) t 6(1 t)t t By smple computaton we have B(0) R (0 ) = P 0; B(1) R (1 ) = P ; 1 1 B( ) (P0 6P1 6P P ) B( ) P (P0 P1 P P ) 14 P0 P P Where w w 1 w 1w we have And for R( ) - P B( ) P 14 8(1 )( ) 14 8(1 )( ) 1 1 R( ) - P = a B( ) - P ; a Where 14 8(1 )( ) b 14 8(1 )( ) P0 P P Equaton (6) holds. These equatons show that Cubc WAT- Ratonal Bezer Curve can be made closer to IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 694

5 the control polygon by alterng the values of shape parameters. Corollary.1: The quadratc trgonometrc Bézer Curves wth tenson parameter s closer to the control polygon that the cubc Bezer curve f only f Corollary.: when a = b = 1 the Cubc WAT- Ratonal Bezer Curve can be closer to the cubc 1 1 B( ) R ( ); Ratonal Bézer Curve.e. Fgure 4. show the relatonshp among the Cubc WAT- Ratonal Bezer Curve (yellow lne) the cubc Ratonal Bézer Curve wth shape parameter (blue lne) the cubc Bézer Curve ( green lne). Fgure 4. Relatonshp among the Cubc WAT- Ratonal Bezer Curve the cubc WAT- Bézer Curve wth shape parameter the cubc Bézer Curve. 5. JOINTING OF WAT- RATIONAL BÉZIER CURVES Suppose there are two segment of WAT- Ratonal Cubc Bezer curves Pw b (t 1) R(t 0 1) wb (t 1) 0 Qwb (t ) Q(t 0 ) wb (t ) 0 P Q where 0 parameters of R(t 1) Q(t ) are 1 respectvely. ; To acheve G 1 contnuty of the two curve segments t s requred that not only the last control pont of R(t ) Q(t ) 1 the frst control pont of must be the same but also the drecton of the frst order dervatve at jontng pont should be the same namely ' ' R 1 kq ( 0) ;(k 1) Substtutng Eq. () nto the above equtaton one can get w w 1 P P 1 k Q 1 Q0 w w0 ww 1 k ww Let 0 1 substtutng t nto the above equtaton then P P Q Q ww 1 ww Especally for k=1 namely 0 1 the frst order dervatve of two segment of curves s equal.thus G 1 contnuty has transformed nto C 1 contnuty. Then we can get followng theorem 4. Theorem 4 If P P Q O Q 1 s collnear have the same drectons.e. (P P ) = δ (Q 1 Q 0 ) (δ > 0) (7) Then curves of P(t) Q(t) wll reach G 1 contnuty at a jontng pont when ww 1 ww 0 1 they wll get C 1 contnuty. Then we wll dscuss contnuty condtons of G when λ 1 = λ =1. Frst we ll dscuss condtons of G contnuty whch s requred to have common curvature namely ' '' R 1 R 1 ' '' Q 0 Q 0 '' '' R 1 Q 0 (8) Let λ 1 = λ =1 second dervatves of two segments of curves can be get '' 1 R (1 1) w w P P 1 1 ww w P P w (9) '' 1 Q (0 ) 6 1 w w P P 1 1 w0w1 w1 P P w 0 Substtutng Eq. () (9) nto Eq. (8) smplfyng t then IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 695

6 ( P P ) ( P P1 ) ( Q1 Q0) ( Q Q1) P P Q1 Q0 (10) Substtutng Eq. (7) nto the above equtaton one can get h 1 = δ h where h 1 s the dstance from P 1 to P P h s the dstance from Q to Q 0 Q 1. Hence we can get theorem 5. Theorem 5 Let parameters λ 1 λ are all equal one f they satsfy Eq. (7) (10) fve ponts P 1 P P Q 1 Q are coplanar P 1 Q are n the same sde of the common tangent then jontng of curves R(t 1) Q(t ) reach G contnuty. Proposton 4. Let P 0 P 1 P P be four properly chosen control ponts such that ( ) ( ) ( ) ( )( ) Then the correspondng WAT-Ratonal Bézer curve wth the weght parameters λ=0 t [0 1] represents an arc of a helx. Proof: Substtutng P 0 P 1 P P nto () yelds the coordnates of the WAT- Ratonal Bézer curve x (t ) = a cos πt y (t ) = a sn πt z (t ) = bt whch s parameter equaton of a helx see Fgure APPLICATIONS OF WAT- RATIONAL BÉZIER CURVES AND SURFACES Proposton 4.1 Let P 0 P 1 P P be four control ponts. By proper selecton of coordnates ther coordnates can be wrtten n the form ( ) ( ) ( ) ( ) ( ) Then the correspondng WAT- Ratonal Bézer curve wth the weght parameters λ = 0 t [0 1] represents an arc of cyclod. Proof: If we take P 0 P 1 P P nto () then the coordnates of the WAT- Ratonal Bézer curve are a 1 cos t x t a t sn t y t It s a cyclod n parametrc form see Fgure 6. Fg.7 The Representaton of Helx Wth WAT- Ratonal Bézer Curve Proposton 4. Gven the followng four control ponts P0 = (00)P1 = P = (a πb)p = (a 0) (ab 0).Then the correspondng WAT- Ratonal Bézer curve wth the weght parameters λ = 0 t [0 1] represents a segment of sne curve. Proof: Substtutng P0 P1 P P nto () we get the coordnates of the WAT- Ratonal Bézer curve x (t )= at y (t )= b sn πt whch mples that the correspondng WAT- Ratonal Bézer curve represents a segment of sne curve see Fgure5. Fg. 6 The Representaton of Cyclod Wth WAT- RATIONAL Bézer Curve FIG.8 The Representaton of Sne Curve wth WAT- Ratonal Bézer Curves IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 696

7 7. CONCLUSIONS In ths paper the WAT-Ratonal Bézer curves have the smlar propertes that cubc Bézer curves have. The jontng of two peces of curves can reach G C contnuty under the approprate condtons. The gven curves can represent some specal transcendental curves. What s more the paths of the curves are lnear the WAT- Ratonal Bézer curves have more advantages n shape adjus tng than that C- Bézer curves. Both ratonal methods (NURBS or Ratonal Bézer curves) WAT- Ratonal Bézer curves can deal wth both free form curves the most mportant analytcal shapes for the engneerng. However WAT- Ratonal Bézer curves are smpler n structure more stable n calculaton. The weght parameters of WAT- Ratonal Bézer curves have geometrc meanng are easer to determne than the ratonal weghts n ratonal methods. Furthermore some complex surfaces can be constructed by these basc surfaces exactly. Whle the method of tradtonal Cubc Bézer curves needs jonng wth many patches of surface n order to satsfy the precson of users for desgnng. Therefore the method presented by ths paper can rase the effcent of consttutng surfaces precson of representaton n a large extent Meanwhle WAT- Ratonal Bézer curves can represent the helx the cyclod precsely but NURBS cannot. Therefore WAT- Ratonal Bézer curves would be useful for engneerng. REFERENCES [5] Farn G. Curves Surfaces for Computer Aded Geometrc Desgn 4th ed. Academc Press San Dego 1997 CA. [6] Hoffmann M. L Y. J. Wang G.Z. Paths of C-Bézer C-B-splne curves Computer Aded Geometrc Desgn vol.pp May 006. [7] Lan YANG Juncheng L. A class of Quas- Quartc Trgonometrc curve surfaces Journal of nformaton Computng Scence Vol. 7 No pp [8] Lu Huayong Extenson And Applcaton Of Cubc Trgonometrc Polynomal Bezer Curve Journal of nformaton Computatonal Scence 9:10 (01) [9] Xe Jn Lu Xaoyan Lu Zh The WAT Bezer Curves Its Applcatons Journal of Modern Mathematcs Fronter Vol. Issue 1 March 01. [10] Zhang J.W. C-Bézer Curves Surfaces Graphcal Models Image Processng vol. 61 pp.: 15 Aprl [1] Dube M. sharma R. Quartc trgonometrc Bézer curve wth a shape parameter. Internatonal Journal of Mathematcs Computer Applcatons Research01 vol. Issue(89-96): [] Dube M.Twar P. Convexty preservng C ratonal quartc trgonometrc splne.aip Conference Proceedng of ICNAAM : [] Dube M. sharma R. Shape Preservng Splne curves Surfaces LAMBERT Academc Publcaton 016 [4] E. Manar T M. Pea J. Sanchez-Reyes. Shape preservng alternatves to the ratonal Bézer model. Computer Aded Geometrc Desgn : IJIRT INTERNATIONAL JO URNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 697