Bézier type surfaces. Applied Mathematics & Information Sciences An International Journal. 1. Introduction

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1 Appl. Math. Inf. Sci. 7 No Applied Matheatics & Inforation Sciences An International Journal c 203 NSP Bézier type surfaces Pişcoran Laurian-Ioan Ovidiu T. Pop 2 Bărbosu Dan 3 Technical University of Clu Napoca North University Center of Baia Mare Victoriei Roania 2 National College "Mihai Einescu" Mihai Einescu Street Satu Mare Roania 3 Technical University of Clu Napoca North University Center of Baia Mare Victoriei Roania Received: Jul ; Revised Oct. 202; Accepted Oct Published online: Mar. 203 Abstract: In this paper with the help of the fundaental polynoials fro general operators we construct Bézier-type GBS Bézier-type surfaces which correspond to the given control points. Keywords: Linear positive operators bivariate operators GBS operatorsbézier-type GBS Bézier-type surfaces. Introduction Let N be the set of positive integers N 0 = N 0}. In this section we recall soe notions which we will use in this paper. We consider I R I an interval we shall use the function sets: BI = f f : I R f bounded on I} CI = f f : I R f continuous on I} C B I = BI CI. For any x I let the functions ψ x : I R ψ x t = t x for any t I e 0 : I R e 0 x = for any x I. If I R is a given interval f BI then the first order odulus of soothness of f is the function ω f ; : [0 R defined for any δ 0 by ω f ;δ = sup f x f x : x x I x x δ }. Let I I 2 J J 2 R be intervals EI I 2 FJ J 2 which are subsets of the set of real functions defined on I I 2 respectively J J 2 L : EI I 2 FJ J 2 be a linear positive operator. The operator UL : EI I 2 FI J I 2 J 2 defined for any function f EI I 2 any xy I J I 2 J 2 by UL f xy = L f x + f y f xy 2 is called GBS operator "Generalized Boolean Su" operator associated to the operator L where " " "" st for the first second variable see [2] or [7]. If f EI I 2 xy I I 2 let the functions f x = f x f y = f y : I I 2 R f x st = f xt f y st = f sy for any st I I 2. Then we can consider that f x f y are functions of real variable f x : I 2 R f x t = f xt for any t I 2 f y : I R f y s = f y sy for any s I. Let I I 2 R be given intervals f : I I 2 R be a bounded function. The function ω total f ; : [0 [0 R defined for any δ δ 2 [0 [0 by ω total f ;δ δ 2 =sup f xy f x y : xyx y I I 2 x x δ y y δ 2 } is called the first order odulus of soothness of function f or total odulus of continuity of function f see [2] or [7]. The first order odulus of soothness for bivariate functions has properties siilar to the properties of the first odulus of soothness for univariate functions. If L is a sequence of operators L : EI FJ N for N i N 0 define T i by 3 T i L x = i L ψ i x x 4 for any x I J where EI FJ are subsets of the set of real functions defined on I respectively J. In application we use the fundaental polynoials fro Bernstein Bleiann-Butzer-Hahn operators. Corresponding author: e-ail: ovidiutiberiu@yahoo.co c 203 NSP

2 484 L.I. Pişcoran O.T. Pop D. Bărbosu: Bézier type surfaces... For N let B : C[0] C[0] the Bernstein operators defined for any function f C[0] by B f x = p x f 5 =0 where p x are Bernstein polynoials defined as follows p x = x x 6 for any x [0] any 0...} see [3] or [8]. In 980 G.Bleian P.L.Butzer L.Hahn introduced in [4] a sequence of linear positive operators L L : C B [0 C B [0 defined for any function f C B [0 by L f x = + x =0 x f + 7 for any x [0 any N where C B [0 = f f : [0 R f bounded continuous on [0 }. This class of operators has been intensively studied obtaining various generalizations. One of the ost recent approaches aied at q-calculus see []. 2. Preliinaries For the following constructions the results as well see [7]. In this section let p = for any N or p = for any N siilarly is defined q n n N. Let I I 2 J J 2 R be intervals with I J /0 I 2 J 2 /0. For n N 0... p } N q n } N 0 we consider φ : J R φ x 0 for any x J ψ n : J 2 R ψ n y 0 for any y J 2 the linear positive functionals A : E I R B n : E 2 I 2 R. For n N define the sequences of operators L K n n by L f x = K n gy = p =0 q n =0 φ xa f 8 ψ n yb n g 9 for any f E I g E 2 I 2 x J y J 2 where E I E 2 I 2 are subsets of the set of real functions defined on I respectively I 2. In the following let s N 0 s even. We suppose that the operators L K n n verify the conditions: there exist the sallest α β [ s + 2} such that T L x li α = a x 0 for any x I J T n K n y li n n β = b y for any y I 2 J 2 if we note γ s = ax α s 2l+β2l : l then where l s s 2 α s 2l + β 2l+2 γ s 2 < 0 α s 2l+2 + β 2l γ s 2 < 0 α s 2l+2 + β 2l+2 γ s 4 < 0 }. }} 2 3 In the following we consider the set EI I 2 = f f : I I 2 R f x E 2 I 2 for any x I f y E I for any y I 2 }. For n N let the linear positive functionals A n : EI I 2 R with the property A n x i y l = A x i B n y l 4 for any 0... p } N q n } N 0 il 0...s} x I y I 2. Let n N. The operator Ln defined for any function f EI I 2 any xy J J 2 by L n f xy = p q n =0 =0 φ xψ n ya n f 5 is naed the bivariate operator of LK-type. In the following we consider that for any x I J N T 0 L x = A 0 e 0 = 6 T n0 K n y = B n0 e 0 = 7 for any y I 2 J 2 n N. Fro 6 7 it results iediately that for any x I J N p φ x = 8 =0 q n =0 ψ n y = 9 for any y I 2 J 2 n N α 0 = β 0 = 0. c 203 NSP

3 Appl. Math. Inf. Sci. 7 No / In the following in addition we suppose that α s+2 < α s + 2 β s+2 < β s for any f EI I 2 we have A n f x = B n f x 2 A n f y = A f y 22 A n f = A B n f x = B n A f y 23 for any x I y I p } N q n } N 0 ; n N. Now let UL be the GBS operators associated to the Ln n n operators. If n N then UL n have the for UL n f xy = K n f x y + L f y x L n f xy 24 for any xy I J I 2 J 2 any f EI I 2. Now we recall two results fro [7] which are obtained for s = 0 which we will use in this paper. Theore.Let f : I I 2 R be a bivariate function. If xy I J I 2 J 2 f is continuous in xy then li L f xy = f xy 25 li UL f xy = f xy. 26 Assue that f is continuous on I J I 2 J 2 there exist the intervals K I J K 2 I 2 J 2 such that there exist 0 N a 2 b 2 R depending on K respectively K 2 so that for any N 0 any x K y K 2 we have T 2 L x α 2 T 2 K y β 2 a 2 27 b Then the convergence given in are unifor on K K 2 L f xy f xy 29 +a 2 +b 2 ω total f ; δ 0 δ 0 UL f xy f xy 30 + b 2 ω f x ; + + a 2 ω f y ; 2 β 2 2 α 2 + +a 2 +b 2 ω total f ; δ 0 δ 0 +b 2 ω f x ; + +a 2 ω f y ; + δ 0 δ 0 +a 2 +b 2 ω total f ; δ 0 δ 0 for any xy K K 2 any N 0 where δ 0 = ax β 2 2α 2 2 } 2 α 2 + β Bézier type surfaces Let K K 2 be the intervals fro the Theore. For n N let the nodes x K y n K 2 z n R where 0... p } N q n } N 0. In the following we consider a continuous function on K K 2 f : K K 2 R so that f x ;y n = z n where n N 0... p } N q n } N 0. Definition.Let n N. The point M n = x ;y n ;z n K K 2 R where 0... p } N q n } N 0 is called control point of n order. Definition 2.Let n N. The LK-Bézier surface respectively GBS-Bézier surface of n order which correspond to the control points M n 0... p } N q n } N 0 are defined by B n uv = respectively B n uv = p q n =0 =0 q n =0 p q n =0 =0 p q n =0 =0 φ uψ n vm n 3 ψ n vm n u + φ uψ n vm n p =0 φ uψ n vm n = φ un v u + N v M n. 32 c 203 NSP

4 486 L.I. Pişcoran O.T. Pop D. Bărbosu: Bézier type surfaces... where uv K K 2 M n u = u;y n ;z n u N v = x ;v;z 2 vzn u = f u;y n z 2 v = f x ;v 0... p } N q n } N 0. In the following we consider that A n f u = B n f u = f uy n A n f v = A f v = f x v A n f = f x y n for any uv K K p } N q n } N 0 n N. Then fro one obtains B n uv = L e u;k n e v;l n f uv 33 UB n uv = u;v;ul n f uv 34 for any uv K K p } N q n } N 0 n N. In the exaples fro this paper we have that α 2 = β 2 = γ 0 = exist the constants a 2 b 2 verifying in every application. Taing Theore into account for the construction above the following theore holds. N 0 v = x 0;v; f x 0 ;v = ;v;v N v = x ;v; f x ;v = ;v;v. In the below figure is the graphical representation of the function f which have the following paraetric equation: xuv = u yuv = v zuv = u 2 v where uv [0 [0. Theore 2.The following convergence li B uv = u;v; f uv 35 li UB uv = u;v; f uv 36 are unifor in K K 2 Exists 0 N so that L f uv f uv + a 2 + b 2 ω total f ; UL f uv f uv + b 2 ω f u ; + + a 2 ω f v ; + + a 2 + b 2 ω total f ; for any uv K K 2 any N 0. Next in applications we consider = n = let be the function f : [0 [0 R f uv = u 2 v for any uv [0 [0. Also we tae x 0 = x = y 0 = 0 y = 2 z 00 = 2 z 0 = 6 z 0 = 2 z = 2 then the control points of order are M 00 = ;0; 2 M 0 = ;2; 6 M 0 = ;0;2M = ;2;2. One obtains M 0 u = u;y 0; f u;y 0 = u;0;0 M u = u;y ; f u;y = u;2;2u 2 Application Let K = K 2 = [0] φ u = p u ψ n v = p n v uv [0] n N 0...} 0...n} using the above conditions one obtains: B uv = p 0 up 0 vm 00 + p up 0 vm 0 + p 0 up vm 0 + p up vm using this one obtains: B uv = + 2u;2v; 2 + 4u 4v + 4uv uv [0] The paraetric equations of the above surface are: xuv = + 2u yuv = 2v zuv = 2 + 4u 4v + 4uv where uv [0] the graph of this surface is plotted c 203 NSP

5 Appl. Math. Inf. Sci. 7 No / ψ n v = n +v n v uv [0 n N 0...} 0...n} using the above conditions one obtains: B uv = φ 0 uφ 0 vm 00 +φ uφ 0 vm 0 + φ 0 uφ vm 0 + φ uφ vm using this one obtains: B uv = +u u ; +v 2v ; 2+2u 6v+2uv +u+v. The paraetric equations of the above surface are: xuv = u +u yuv = +v 2v zuv = 2+2u 6v+2uv +u+v where uv [0 the graph of this surface is plotted On the other h one obtains UB uv = p 0 vm 0 u + p vm + u p 0 un 0 v + p un v B uv = u;v;2 4u + 5v 4uv + 2u 2 v uv [0] using this one obtains the paraetric equations of the GBS-surface which are xuv = u yuv = v zuv = 2 4u + 5v 4uv + 2u 2 v where uv [0] the graph of this surface is plotted The GBS-surface is: UB uv = φ 0 vm 0 u + φ vm + u φ 0 un 0 v + φ un v B uv = = u;v; 2u3 v + 2u 2 v + 2uv 2 + v 2 + 7v uv + 2 2u + u + v Application 2 Let K = K 2 = [0 φ u = +u u xuv = u yuv = v zuv = 2u3 v+2u 2 v+2uv 2 +v 2 +7v uv+2 2u +u+v where uv [0 the graph of this surface is plotted c 203 NSP

6 488 L.I. Pişcoran O.T. Pop D. Bărbosu: Bézier type surfaces... Application 3 Let K = [0]K 2 = [0 φ u = p u ψ n v = n +v n v u [0]v [0 n N 0...} 0...n} then: B uv = φ 0 uψ 0 vm 00 +φ uψ 0 vm 0 + φ 0 uψ vm 0 + φ uψ vm using this one obtains: B uv = 2u ; +v 2v ; 2+4u 6v+8uv +v UB uv = ψ 0 vm 0 u + ψ vm + u φ 0 un 0 v + φ un v B uv = = u;v; 2u2 v 8uv 4u + v 2 + 7v v The Bézier surfaces GBS-Bézier surfaces fro this application are given paraetrically by respectively xuv = 2u yuv = +v 2v zuv = 2+4u 6v+8uv +v xuv = u yuv = v zuv = 2u2 v 8uv 4u+v 2 +7v+2 +v where u [0v [0 the graphs of these surfaces are plotted References [] Agratini O. Nowa G. On a generalization of Bleiann Butzer Hahn operators based on q-integers Matheatical Coputer Modelling Vol Issues [2] Badea C. Cottin C. Korovin-type Theores for Generalized Boolean Su Operators Colloquia Matheatica Societatis János Bolyai 58 Approxiation Theory Kecsceét Hungary [3] Bernstein S. N. Déonstration du théorèe de Weierstrass fondée sur le calcul de probabilités Coun. Soc. Math. Kharow [4] Bleiann G. Butzer P. L. Hahn L. A. Bernstein-type operator approxiating continuous functions on the sei-axis Indag. Math c 203 NSP

7 Appl. Math. Inf. Sci. 7 No / [5] Ditzian Z. Toti V. Moduli of Soothness Springer Verlag Berlin 987. [6] Pop O. T. Barbosu D. Pi scoran L. Bézier type curves Creative Math. & Inf No [7] Pop O. T. The approxiation of bivariate functions by bivariate operators GBS operators Rev. Anal. Nu. Theor. Approx No [8] Stancu D. D. Coan Gh. Agratini O. Trîbi ta s R. Analiza nuerica s i teoria aproxiarii I Presa Universitara Clueana Clu-Napoca 200 in Roanian. [9] Tian A. F. Theory of Approxiation of Functions of Real Variable New Yor: Macillan Co. 963 MR22# Pi scoran Laurian Ioan is a lecturer at Technical University of Clu-Napoca North University Center of Baia Mare Roania. He obtained his PhD. at Babe s-bolyai University of Clu-Napoca Roania. He is an active researcher has a 2 year teaching experience. He has ore than 25 research articles soe of the published in reputed international ournals. He is editor in chief at two International Geoetry Journals also a reviewer at Matheatical Reviews Zentralblatt fur Matheati. His interest topics are in Geoetry: curves surfaces Rieannian geoetry Finsler geoetry hyperbolic geoetry Euclidean geoetry proective geoetry noncoutative geoetry. Ovidiu T. Pop is a professor of atheatics in the Departent of Matheatics of the "Mihai Einescu" National College in Satu Mare Roania. In 2006 he received the Ph.D. in Matheatics at Babe s-bolyai University of Clu-Napoca. His research filds include the theory of linear positive operators. He has published 3 papers out of which 80 are research papers 0 having been published in ISI ournals. Barbosu Dan Professor Barbosu Dan was born in in BaiaMare Roania. He was a very good student of the Faculty of Matheatics Inforatics of the University "Babe s-bolyai" Clu-Napoca where he obtined his diploa as teacher professor in atheatics 979. In 980 he obtained the degree of aster in "Nuerical Analysis" at the sae university. In the period he was teacher of atheatics in BaiaMareat High School.Nr.6. Starting 990 he wored at North-University of Baia-Mare as assistant professor lecturer finally as associate professor at the sae university. This period he published ore as 00 papers regarding the approxiation by linear operators quadratures cubatures based on linear operators in: Carpathian Journal of Matheatics Inforatics Baia-Mare Misolc Matheatical Notes Hungary Deonstratio MatheaticaPoll Matheatica Balcania Bulgaria Facta Math.NisSerbia Annals of "Ovidius" Univesity Constantza Roania Annals of the University of CraiovaRoaniaetc. c 203 NSP