Summary of Curve Smoothing Technology Development

Transcription

1 RESEARCH ARTICLE Summary of Curve Smoothg Techology Developmet Wu Yze,ZhagXu,Jag Mgyag (College of Mechacal Egeerg, Shagha Uversty of Egeerg Scece, Shagha, Cha) Abstract: Wth the cotuous developmet of computer techology, the applcato of free curve the desg ad maufacture of moder dustral products s more ad more wdely used.however, due to errors the desg ad measuremet process, the smoothess of the curve ofte ca ot meet the desg ad maufacturg requremets.the smoothess of the curve affects ot oly the appearace of the product, but also the dffculty of maufacturg techology ad the mechacal propertes of the products. It s a mportat symbol to evaluate the qualty of products. Dscussed the terpolato of the curve ad the approxmate costructo method, the smoothg crtero ad the smoothg method ad troduced the relatoshp betwee smooth curve ad smooth surface. Keywords Curve, farg, Geometrc modelg OPEN ACCESS 1. INTRODUCTION Computer Aded Desg (CAD) ad Computer Aded Maufacturg (CAM) techology orgated the avato dustry, due to the complex arcraft shape, wth a large umber of free-form curves ad surfaces, therefore, CAD / CAM techology from the very begg wth the freeform surface modelg techology are closely lked. Curved surface modelg techology was establshed by Coos, Bezer ad other masters the 196s [1]. After years of research ad developmet, curve shape of surface formed by NURBS curve characterstcs of parameterzed desg ad mplct algebrac curves ad sad the two types of method as the ma body, by meas of terpolato, approxmato, fttg for the skeleto geometry theory system [2]. Alog wth the developmet of the dustry, because of the NURBS method ca accurately sad secod rule curve surface, through the power factor s easy to cotrol ad mplemetato, ad ca drectly promote such outstadg advatages thkg space, 1991, the teratoal orgazato for stadardzato (ISO) defe the NURBS as the geometry of the dustral products oly mathematcal method.sce the, NURBS method has become the most mportat foudato the tred of curve surface modelg techology [3]. 2. CURVE INTERPOLATIONANDAPPROXIMATION CONSTRUCTION MRTHOD Iterpolato occurs frst the feld of umercal aalyss ad s usually terpreted to geerate ew data through kow data.i a curved structure, t s gve that some data pots geerate a smooth curve.approxmatos also geerate ew data through kow data.the dfferece betwee the two s that the curve produced by the terpolato method passes through all the pots gve, callg these pots data pots. The curves geerated by the approxmato method do ot ecessarly go through all the pots. These pots are called cotrol pots[4]. ISSN: Page 7

2 2.1.The Bezer Curve I the early 196s, Bezer ad De Casteljau proposed the same curve costructo method by dfferet algorthms--bezer curve [5]. Bezer method s polyomal P ad the terpolato ad approxmato.now we have a gve pot ( ) umber s + 1,f gve + 1weghted coeffcet, meet the followg codtos W = = 1, the pot Q= = W P, ca thk of t as a lear combato or the pot of weghted average. If W s a fucto of parameter t, that's what we're talkg about W = W ( t), t [ a, b], usually, the value of t s [,1], ad the P( t) = W ( t) P, t represets a parametrc curve, ad at ths momet W ( t ) s base fucto or bledg fucto, the polygo that coects the cotrol pot P s called the cotrol polygo. To costruct a smooth curve, Bezer chose Berste polyomal as the bass fucto, t s B ( t) ( ) t ( 1 t ) the umber of pots. Ths equato ( ) of the curve. It ca be see that ( ) ISSN: Page 8 =, ths fucto, s = B t = 1 guaratees the affe varace B t s a polyomal ad ts frequecy s. Bezer curve terpolato P P, they are Attracted by other cotrol pots. Ths Bezer curve approxmates the curve. You ca cotrol the shape of the curve by movg these pots, or you ca add ad remove pots to chage the shape. You ca also costruct terpolated Bezer curves. Now we have a gve pot Q, choose t ad ts umber s 1 P t = Q, the ( ) + whch make ( ) form of a matrx s expressed as: B ( t ) B1 ( t ) L B ( t ) P Q B ( t1 ) B1 ( t1 ) L B ( t1 ) P1 Q1 = (1) M M O M M M B ( t ) B1 ( t ) B ( t ) L P Q The equato (1)s sad as MP = Q, Let's multply both sdes of ths equato by 1 M ad obtaed obtaed 1 P M Q =. The, the drawg of the curve s obtaed by the Q s obtaed. P, ad the Bezer curve of terpolato 2.2.The B-sple Curve The b-sple costructo method of the curve was proposed the 194s, ad the breakthrough was developed the 197s, such as Rch Resefeld, De Boor et al[5]. B-sple techology has overcome the dsadvatages of Bezer techology, has the advatages of local cotrol, ad has hgh order cotuty. Smlarly, the B sple curve s the approxmato curve. The b-sple curve has the same expresso as the Bezer curve, but the selecto prcple of the base fucto s dfferet. The b-sple curve troduces the ode vector, whch s the parameter value terval P ad ts umber s + 1, of the curve.suppose you have a kow pot ( ) the curve P ( t) = = Wk, ( t) P, t [ a, b] bass fucto W ( t ) s a K degree polyomal, 1 k, ( ) k, s called a K-order B-sple curve, f the P t s a pecewse polyomal fucto ad t has a cotuous dervatve at the jucto utl k 1.

3 Whle k =, the B-sple curve degeerates to a Bezer curve, so the Bezer curve W t s a specal case of the B-sple curve. The structure of the base fucto ( ) adopts the cox-de Boor recursve algorthm [6]. The B-sple curve troduces the ode vector, whch s the parameter value rage of the curve.the pot vector s a o-decreasg sequece of real Numbers ( t, t1, L, t m ), where t s called a ode. The dstrbuto of odes produces dfferet B-sples ad the uform B-sple has more flexble cotrol, that s, cotrol pots ca be mapulated, or odes ca be moved or serted. However, a B-sple curve ca ot represet a coc curve because the fucto of the coc curve s a ratoal polyomal. Therefore, ratoal B-sple s troduced. Geerally, use pot homogeeous coordate to uderstad ratoal B-sple. Three-dmesoal pot w,, P w x, w y, w z, w. P ( x y z ) ca use homogeeous coordates as ( ) k, Substtutg the homogeeous coordates to the B-sple curve's parametrc equato: Wk, ( t) w x = w x, ( ) Wk t w y w w w y = P ( t) = Wk, ( t ) P Wk, ( t) = = = = w z Wk, ( t) w z w = Wk, ( t) w = The frst three compoets wth the fourth compoet to three-dmesoal form s: Wk, ( t) w x / Wk, ( t) w = = ww k, ( t) P ( t) = Wk, ( t) w y / Wk, ( t) w = p = k, ( ) j j k, j ( ) R t P = = = = w W t = Wk, ( t ) w z / Wk, ( t) w = = R t. Ratoal B-sple You ca see the ratoal polyomal of bass fucto ( ) ca accurately represet the coc curve [5]. 3. CURFACESMOOTHING The smoothess of the curve maly cludes two aspects: the smooth ad the smooth method [3][7]. 3.1.Curve Smoothg Crtero It s geerally cosdered that the followg codtos should be cluded the smoothess crtero:secod order cotuous curve; there are o sgulartes or excess flecto pots;the curvature chage s relatvely uform (some curves are ot oly due to the sharp crease or decrease of curvature, but also the curve s ot smooth);stra eergy s small. k, ISSN: Page 9

4 3.2.Curve Smoothg Method ad Aalyss The smoothg method s to make a certa mathematcal treatmet of ot oly the smooth curve to acheve the smooth (or the smoothess). I the exstg lght smoothg method, accordg to the lght smoothg crtero ad the umber of modfed value pots, t s dvded to local smoothg method [8-1] ad the whole smoothg method [11-13]. Local smoothg method, such as roud rate method, legal persos ad other are modfed method for choosg the base samples, s uder the assumpto that most values are good or better, o the bass of the morty ot oly alog the area of the type value pots (kow as "bad pots") ameded pck out oe by oe.usually, ths kd of method has strog local modfcato ablty ad fast calculato speed, but t s ot deal whe the "bad pot" has more tme.the commo feature of local smoothg method s that the curve ca be adjusted locally accordg to the specfc requremets of the curve desg. The dsadvatage s that the algorthm ca oly adjust oe cotrol vertex at a tme.local algorthm sutable for some smple curve farg, complex curve to the large amout of data usg local farg method ofte eed to repeatedly adjust curve, ot oly the computato, ad dffcult to guaratee the overall effect of the curve. The whole smoothg method, such as the least square method, eergy method ad sprgback method, s the mmzato problem of the trasformato of the smooth problem to the objectve fucto, ad the basc dea s the same.the target fucto cotas the devato of the type value pot ad the weghted average of the two parts. Geerally, ths kd of method s deal for overall smoothg effect, but the calculato s large ad the covergece speed s slow.the eergy method s a wdely used curvlear smoothg method. The eergy method takes to accout the mor crtero of stra eergy, ad the uform crtero of curvature chage s ot cosdered.i addto, usg the eergy method to smooth the lght, o matter what the geometrcal shape of the smooth, the smooth effect always teds to the lear chage, so some cases the smooth effect s ot satsfactory. At preset, most CAD/CAM systems adopt the parameter B-sple curve techology, ad the most commo method egeerg practce s to terpolate the curve by mportg or creatg the type value pot.the terpolato curve s strctly based o the gve value pot, ad the premse of the terpolato algorthm s accurate.however, due to the measuremet error ad other reasos, t s mpossble to get the exact type pot.i addto, for some have more tha oe area of the curve, ot oly because of the desg tself, some areas ca ot accordg to rule of farg farg, otherwse t wll affect the product's desg tet. 4. THE RELATIONSHIP BETWEEN SMOOTH SURFCE AND SMOOTH SURFACE The process of reverse modelg s usually composed of the pot structure, the surface s coected by the trasto surface, ad the surface recostructo s completed. Therefore, the smooth desg of the curve s the bass of smooth surface desg. Class A surface s A proper ou the feld of automoble, frst put forward by the Frech dassault compay, t should ot oly coform to the requremets of the cotuty of mathematcs, ad complace requremets, to meet the requremets of aerodyamcs ad lght cotrol pots should be evely arraged o the surface [14].It ISSN: Page 1

5 s dffcult to recostruct the curvature cotuous surface wth the pot cloud to costruct the curvature cotuous surface.automotve coverg parts A level surface to acheve the curvature, must be based o the curvature of curve recostructo surface curvature chage eve o redudat flecto pot ad the curvature error wth.5 mm hard to reach A level surface. Surface smoothess s a mportat crtero for evaluatg A-grade surface, ad the surface smoothg methods clude curvature detecto method, eergy aalyss, wavelet aalyss, geetc algorthm smoothg ad llumato detecto.for A sgle surface, the A class surface requres t to be cocave. Requremets the dustral area A-class surface must meet the followg requremets: (1) the order of the surface the two drectos should be betwee 3 ad 7 tmes, ad the maxmum should ot be greater tha 9 tmes. (2) the adjacet surface flm satsfes G 2 cotuous, ad the specal requremet satsfes G 3. (3) large feature curved plates are sgle covex. 5. CONCLUSIONS The developmet of curve smoothg techology provdes strog support for moder maufacturg dustry.the modelg techque s based o the tradtoal curve terpolato, approxmato, fttg ad exteso to varous optcal smoothg techques.the CAD dustry has also rapdly appled ew techologes ad theores to dustry applcatos, ad the mportace of vsble lght techology to moder dustry.lookg at the developmet of geometrc modelg techology, t s the key to promote the whole CAGD research to fd better curve smoothg techology. REFERENCES [1] Bohem, W, Far, G ad Kahma, J. A survey of curve ad surface methods CAGD, Computer Aded Geometrc Desg., vol. 1, pp. 1-6, [2] Yoggag Lv. Study o uform sple CAGD free curve surface modelg, Doctoral dssertato of Zhejag Uversty., 22. [3] Xgxog Zhu. Free curve surface modelg techology [M]. Bejg: Bejg Uversty of Aeroautcs ad Astroautcs press, 2. [4] Juhua She,Guzhe Sh. A revew o the developmet of free curve surface modelg[j]. Iformato Techology., vol. 3, pp , 213. [5] Davd Salomo. Curves ad surfaces for computer graphcs[m]. Sprg New York Ic., 25. [6] Carl De Boor. O calculato wth B-sples[J]. Joe Approxmato theory., vol. 6, pp. 5-62, [7] Fazhog Sh. Computer Aded Geometrc Desg ad No-Uform Ratoal B-sple. Bejg: Bejg Uversty of Aeroautcs ad Astroautcs press., [8] Xaopg Log. The local eergy optmal method ad the smooth surface of the curved surface. Joural of computer aded desg ad graphcs., vol. 14, pp , 22. ISSN: Page 11

6 [9] Sapds N, Fae G. Automatc farg algorthm for B-sple curves. Computer Aded Desg., vol. 22, pp , 199. [1] Polakoff J F. A mproved algorthm for automatc farg of o-uform parametrc cubc sples. Computer Aded Desg., vol. 28, pp , [11] Welag Luo, Xua Yag, Shumg Zhe. The costrat smoothg algorthm of B-sple curves. Joural of Zhejag Uversty.,vol. 31, pp , 24. [12] Lg Jg, Pg X, Rogx Tag. The curve surface s smooth wth the deformato model. Joural of Software.,vol. 9, pp , [13] Pgouaks K G. Farg methods of plaar ad space curves uder desg costrats applcato computer aded shp desg. Athes:Natoal Techcal Uversty of Athes., [14] Yabg Cao, Le Wag, Sfa Zhe, et al. The desg of A pot cloud the desg of the outer cover of the vehcle [J]. Mechacal desg ad maufacturg.,vol. 2, pp , 27. ISSN: Page 12