Curves - Foundation of Free-form Surfaces
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1 //9 Cres - Fodato of Free-form Srfaces Why Stdy Cres? Cres are the bascs for srfaces Whe asked to modfy a partclar etty o a CAD system, kowledge of the ettes ca crease yor prodctty Uderstad how the math presetato of aros cre ettes relates to a ser terface Uderstad what s mpossble ad whch way ca be more effcet whe creatg or modfyg a etty
2 //9 Why Not Smply Use a ot Matrx to Represet a Cre? Storage sse ad lmted resolto Comptato ad trasformato Dffcltes calclatg the tersectos or cres ad physcal propertes of objects Dffcltes desg (e.g. cotrol shapes of a exstg object oor srface fsh of mafactred parts Adatages of Aalytcal Represetato for Geometrc Ettes A few parameters to store Desgers kow the effect of data pots o cre behaor, cotrol, cotty, ad cratre Facltate calclatos of tersectos, object propertes, etc.
3 //9 Aalytc Cres s. Sythetc Cres Aalytc Cres are pots, les, arcs ad crcles, fllets ad chamfers, ad cocs (ellpses, parabolas, ad hyperbolas Sythetc cres clde aros types of sples (cbc sple, B-sple sple, Beta-sple ad Bezer cres. Cred Srfaces I CAD, We wat to fd a math form for represetg cred srfaces, that : (a look ce (smooth cotors (b s easy to maplate ad mafactre (c follows prescrbed shape (arfol desg To stdy the cred srface, we eed to start from cres.
4 //9 arametrc Represetato * a cre * a srface T ( = [ (, (, ( ] (, x(,, y (,, z (, x y z z =cost = T -- p(x,y,z Umax =cost Um -- p( y x We ca represet ay fctos of cre (cred srface sg parametrc eqato. ( C ( C ( C Srface fttg sg dscrete data 4
5 //9 arametrc Represetato of Les How s a le eqato coerted by the CAD/CAM software to the le database? How are the mathematcal eqato correlated to ser commads to geerate a le? z, = -, = = ( - - = ( - x y = ( -, Les z, = -, = x y = ( -, x = x ( x x y = y ( y y z = z ( z z 5
6 //9 Crcle Represetato (No-parametrc x y = (a x = (parameter y = poor defto sqare root complcated to compte Crcle Represetato π/ π/8 π/4 (b x = cos y = s better defto tha (a bt stll slow π/8 6
7 //9 Crcle Represetato Recrse approach x y x x y = = r cos r s θ θ -- = r cos( θ d θ = r cos θ cos d θ r s = x cos d θ y s d θ = y cos d θ x s d θ -- θ s d θ Obserato: cres are represeted by a seres of le-segmets Smlarly all coc sectos ca be represeted. Ellpse x = xo A cosθθ y = yo Bsθ z = zo θ π Th t th th d th R t t The compter ses the same method as the Represetato of crcle to redce the amot of calclato. 7
8 //9 arabola x = xo A y = yo A z = zo arametrc Represetato of Sythetc Cres Aalytccresaresally ot sffcet to meet geometrc desg reqremets of mechacal parts. May prodcts eed free-form, or sythetc cred srfaces. Examples: car bodes, shp hlls, arplae fselage ad wgs, propeller blades, shoe soles, ad bottles The eed for sythetc cres desg arses o occasos: whe a cre s represeted by a collecto of measred data pots ad (geerato whe a cre mst chage to meet ew desg reqremets. (modfcato 8
9 //9 The Order of Cotty The order of cotty s a term sally sed to measre the degree of cotos derates (C, C, C. y y = = = = = = x x Smplest Case Lear Segmet y = a a x Hgh order polyomal may lead to rpples y = a a x... a x Sples Ideal Order Sples a mechacal beam wth bedg deflectos, or a smooth cre der mltple costrats. ( ( ( (4 y x y y ( x = R( x ( x M = EI a b EI 6 ax b = EI ( x = x x c x d Cbc Sple 9
10 //9 Hermte Cbc Sples ( = [ x(, y( z( ] T, ( ( ( x = cx cx c x cx y = c y c y c y c y z = cz cz c z cz r = Cbc Sple T ( = ( ( ( = ( p x y z C = C C = C T [ U ][ C] C 4 = coeffcets to be determed Two Ed ots Hermte Cbc Sples = C = C = r, = C C C = C = = C = C C C C = = C C C Bodary Codtos: Locato of the two ed pots ad ther slopes = p r C C C p -- r 4 eqatos from two cotrol pots = p r -- p r
11 //9 Hermte Cbc Sples C = ( ( C C C = = = r C C C C C = = r kows ad eqatos C C C C C = ( ( ( ( ( r = All parameters ca be determed Hermte Cbc Sples ( ( ( ( ( r = ( ( [ ] = Based o: Locato of the two ed pots ad ther slopes [ ] = V M U H T r ( 4 ( 6 6 ( 6 (6 ( = r
12 //9 Jog Cbc Sple Segmets (= (= m -- m (= r (= m, r r mm m r r r r r r r r m( =, m( = ; m ( = m, m ( = m r r r m ( = m ( = m, m ; r r m( = m (; r r ( = ( m m cre segmets & 4 kows 6 eqs. from frst pot, 6 eqs. from last pot, 4 eqs. from jog pot Go throgh ceter pot, hae same st ad d order derates Jog Cbc Sple Segmets r ( = (6 6 ( 6 6 ( 4 ( r = ( 6 ( 6 (6 4 (6 r r ( = = ( = r ( = = r ( = = 6 6 4
13 //9 Qestos o Cbc Sples What are the cotrol parameters to chage the shape of a cbc sple? What f I wat to chage a local cratre? Is there ay way I ca crease the order of cotty o a cbc sple? Ca I mproe the order of cotty by addg more pots? Dsadatages of Cbc Sples The order of the cre s always costat regardless of the mber of data pots. I order to crease the flexblty of the cre, more pots mst be proded, ths creatg more sple segmets whch are stll of cbc order. The cotrol of the cre s throgh the chage of the postos of data pots or the ed slope chage. The global cotrol characterstcs s ot tte.
14 //9 Bezer Cre. Bezer of the Frech atomoble compay of Realt frst trodced the Bezer cre. A system for desgg sclptred srfaces of atomoble bodes (based o the Bezer cre - passes p ad p, the two ed pots. - has ed pot derates: r r r r r r p p = ( pp pp ; p p= = ( p( p pp = ( - ses a ector of cotrol pots, represetg the ertces of a characterstc polygo. p ( = p B, ( = Math Expresso p r p p p r r p p p r p r p p p r p r p p p r segmet(each polygo ertces (each polygo ad mber of cotrol pots [, ] 4
15 //9 Berste olyomal B B, ( =!! (! (, ( s a fcto of the mber of cre segmets,, ad.!!!! = = =!(!!!!!!! A Example: If =, the = ertces -- p r -- p r p -- p p r!!!! = = =!(!!!!!!! p ( = p B ( =, B, ( =!! (! ( r r r r p ( = ( p ( p p r r r r p( = ( p ( p p r r r r r p( = p p ( = ( p p r r r r r p( = p p ( = ( p p 5
16 //9 The order of Bezer cre s a fcto of the mber of cotrol pots. For cotrol pots (= always prodce a cbc Bezer cre. p p p p p p p p p p p p p p p p p p p p p p p p p p p p p,p p p p,p p 6
17 //9 A Example The coordates of for cotrol pots relate to a crret WCS are ge by [ ] T, = [ ] T, = [ ] T,& [ ] T = = Fd the eqato of the resltg Bezer cre. Also fd pots o cre for =,,,,& 4 4 y? x 7
18 //9 e.g. Solto ( = B, B, B, B, B, ( =!! (! ( ( ( ( = ( Sbstttg the ales to hs eqato ges [ ] T ( = = = = = = [ ] T [.5.75 ] T = = [ ] T =, ¼, ½, ¾, ( = = [ ] T (, (, =/ - cotrol pots,,,, & 4, - pots o cre, ( =/4 =/4 (, (, 8
19 //9 Improemets of Bezer Cre Oer the Cbc Sple The shape of Bezer cre s cotrolled oly by ts defg pots (cotrol pots. Frst derates are ot sed the cre deelopmet as the cbc sple. The order or the degree of the Bezer cre s arable ad s related to the mber of pots defg t; pots defe a th degree cre. Ths s ot the case for cbc sples where the degree s always cbc for a sple segmet. The Bezer cre s smoother tha the cbc sples becase t has hgher-order derates. B-Sple A Geeralzato from Bezer Cre Better local cotrol Degree of resltg cre s depedet to the mber of cotrol pots. 9
20 //9 Math Represetato ( =, k ( max N,,..., = (k- degree of polyomal wth ( cotrol pots cotrol pots. N, k ( B-sple fcto (to be calclated a recrse form N k, N ( N ( ( = ( ( k,, k k k k arametrc Kots N ( N ( Nk, ( = ( ( k,, k k k k : parametrc kots (or kot ales, for a ope cre B-sple: j j j < k = j k k j k j > where, j k, ths f a cre wth (k- degree ad ( cotrol pots s to be deeloped, (k kots the are reqred wth max = k
21 //9 Kot Vale Calclato N k, N ( N ( ( = ( ( k,, k k k k = ; 4 cotrol pots k = 4; 4-= cbc polyomal j k= 7 creases wder base k creases wder & taller Calclato of Nk, ( Fcto N, k ( =, k ( max N = N ( = (, k k ( ( k N, k k < N, = = mx a ad ad = otherwse ( =
22 //9 Role of B-Sple Fcto (Calclatg Weghts N from Nearby Cotrol ots k, ( Nk, ( Nk, ( Lear k= Nk, (.5.75 k=.5 Qadratc 4 4 / /6 Cbc k=4 4 ropertes of B-Sple Nmber of cotrol pots depedet of degree of polyomal Lear k= erte x Qadratc B-Sple k= Cbc B-Sple k=4 Forth Order B-Sple k=5 ertex ertex ertex =; 4 cotrol pots The hgher the order of the B-Sple, the less the flece the close cotrol pot
23 //9 ropertes of B-Sple B-sple allows better local cotrol. Shape of the cre ca be adjsted by mog the cotrol pots. Local cotrol: a cotrol pot oly fleces k segmets ropertes of B-Sple Repeated ales of a cotrol pot ca pll a B-sple cre forward to ertex. ( Iteracte cre cotrol Same order polyomal Add more repeated cotrol pots to pll the cre
24 //9 Dfferet order polyomal Bezer Cre A Example Fd the eqato of a cbc B-sple cre defed by the same cotrol pots as the last example. How does the cre compare wth the Bezer cre? 4
25 //9 Example roblem for Fdg the Bezer Cre Example roblem for Fdg the Bezer Cre 5
26 //9 Fdg the B-Sple Cre for the Same Example roblem Vales to be Calclated = ; 4 cotrol pots k = 4; 4-= cbc polyomal j: j k= 7 creases wder base k creases wder & taller 6
27 //9 Calclatg the Kots, j Calclatg N, N < otherwse, = = max ad ad = = 7
28 //9 Calclatg N,k Calclatg N,k 8
29 //9 Reslt cotrol pots: =4 k degree cre: 4-= 4 cotrol pots cbc polyomal No-form Ratoal B-Sple Cre (NURBS Ratoal B-Sple ( = R, ( max = hn k (, k Rk, = h scalar hn, k =, the R, k ( = N, k ( ( ( If h =,t s the represetato of a B-Sple cre. Idstry Stadard Today! 9
30 //9 Deelopmet of NURBS Boeg: Tger System 979 SDRC: Geomod 99 Uersty of Utah: Alpha- 98 Idstry Stadard: IGES, HIGS, DES, y,,, ro/e, etc.
31 //9 Adatages of NURBS Sere as a gee geeralzatos of o-ratoal B-sple forms as well as ratoal ad o-ratoal Bezer cres ad srfaces Offer a commo mathematcal form for represetg both stadard aalytc shapes (cocs, qadratcs, srface of reolto, etc ad free-from cres ad srfaces precsely. B-sples ca oly approxmate coc cres. rode the flexblty to desg a large arety of shapes by sg cotrol pots ad weghts. creasg the weghts has the effect of drawg a cre toward the cotrol pot. Hae a powerfl tool kt (kot serto/refemet/remoal, degree eleato, splttg, etc. Iarat der scalg, rotato, traslato, ad projectos. Reasoably fast ad comptatoally stable. Clear geometrc terpretatos
32 //9 Qck Qestos mber of cotrol pots; & k degree of the cre (polyomal For B-sple, k shold always be less tha or eqal to. T or F? why? What are the two major adatages of B-sple oer Bezer cre real desg? If k creases ts reasoable rage, wll the correspodg B-sple cre moe closer to the cotrol polygo.? What are the reqred ser pts to costrct a Hermte Cbc, Bezer, B-sple ad NURBS cre segmet? By addg more cotrol pots a small rego whe defg a Bezer cre, oe ca maplate the cratre of the cre wthot affectg other cre sectos. T or F.? B-sple cres are detcal to Bezer cres whe k=. T. or F?
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