This may involve sweep, revolution, deformation, expansion and forming joints with other curves.

Transcription

1 5--8 Shapes ae ceated by cves that a sface sch as ooftop of a ca o fselage of a acaft ca be ceated by the moto of cves space a specfed mae. Ths may volve sweep, evolto, defomato, expaso ad fomg jots wth othe cves. The aalytcal popetes of cves ae deved based o assmptothattheeqatoofthecveskowtos. Howeve desg, the egee ceates a shape sg hs magato wthot kowg the eqato of the cve. The compte shold help the desge sytheszg the cve shape so that oe ca eplcate the maged shape ad chage ad fe te the shape accodgly.

2 5--8 The shape of the cve shold be cotolled by placg oly a few mbe of data pots. The cve ths ceated shold behave lke a elastc stg that a desge ca maplate to gve a desed shape. The cve shold be sythetcally composed of polyomal segmets of lowe degee to avod de oscllatos ad mmze comptato tme ad complexty. The cve model shold have affe popetes sg shape depedece fom the coodate fame of efeece. Ths makes t possble to teat the cve model as a eal object space that does ot get dstoted becase of dffeet fames of efeece. Sce eal desg, complex shapes have to be ceated, t s moe stable to jo togethe seveal segmets of cves, flfllg posto, slope ad/o cvate cottes at the jots. Ths the cves models ae developed whch fom smple bldg blocks fo pecg them togethe to ceate a desed shape. Paametc descpto s pefeed ove the mplct o explct foms as t povdes atclate epesetato of cve segmets thee dmesos. A cbc Fegso s segmet s desged lke a achwth two ed pots ad two espectve slopes. LettheedpotsbeP(x,y,z)atadP+(x+,y+,z+)at.Also lettheespectveslopesbet(p,q,)adt+(p+,q+,+). T adt+eedottobe oftmagtdeadmaybewttetems of t vectos t ad t+ as T c.t ad T+ c+.t+ fo some scalas c ad c+. Let s cosde the paametc vaato alog the x coodate that s x () a a a x + a x + x + x...() wthx()x,x()x+adalsodx()/dtpaddx()/dtp+.weget x a x x + a x + a x + a x + a x p a x p + a x + a x + a x

3 5--8 Solvg whch gves a x x a x p a x x p p ax p x + p WheeΔxx+-xadΔpp+-p.Thepolyomaleq.becomes x() x + p + ( x p p ) + ( p x + p ) x() ( + ( ( ( )x + )x+ + + )p + + )p+ x() H ()x H ()x H ()p H ()p +...() Hee H (),,., ae fctos of paamete ad ae temed as Hemte polyomals. They seve as bledg fctos o bass fctos o weghts to combe the ed pot ad slope fomato to geeate the shape. Ithematxfom,eq.()cabeexpessedas x() x x+...() p p+ Smla teatmet ca be employed fo y() ad z() statg wth the cbc fom eq. () to obta aalogos to eq. (). The combed eslt fo()[x(),y(),z()]maybeexpessedas x y z () [ x() y() z() ] x+ y+ z+ p q p + q + + o P () P +...(4) T T +

4 5--8 We may elocate P o P+ o alte the ed tagets T ad T+ both magtde ad decto to effect shape chage. Fo the Fegso segmet gve by eqato (4), we ca compte the fst ad secod devatves by dffeetatg () wth espect to. ths wold help comptg dffeetal popetes lke tagets ad ed cvates whe mposg cotty codtos at the jcto pots. d() T () (6 6) P + ( 6 + 6) P + + ( 4 + ) T + ( ) T + d d () () ( 6) P + ( + 6) P + (6 4) + (6 ) + T T + d Theaboveeqatocabewttematxfomas P 6 6 ( ) P T T + P ( ) P T T+ Theadsofcvatesgveas () () κ () Theb-omalsgveas The pcpal omal () N () () () T T 4

5 5--8 Fo + data pots P,..,, a eze segmet s defed as the weghted lea combato sg este polyomals as () C( ) P ()P, Fo a composte cve, dvdal segmets eed to be of lowe ode, pefeably cbc. Ths a cbc eze segmet algebac ad matx fom fo data pots P, P, P ad P s gve by () ( ) () ( + () ( P + ( ) P + ( ( 6 )P + + P + P P + P) + (P )P + )P P + ( )P P 6P + P) + ( P + P) + P + 5

6 5--8 Theeqatomatxfomswtteas () 6 P P P P Itmaybeotedthat thecvewllotpassthoghthepots PadP. To chage the shape of the cve, se ca elocate ay of the cotol pots. I Fegso's segmets, the se had to specfy the ed slopes fo a patcla shape, whch s dffclt to specfy. Howeve a eze cve moe o less mmcs the shape of the cotol polyle whch s ease to specfy. No-egatvty: Fo, ae all o egatve ad sce - () as well! () C ( ) ( )!( )! Patto of Uty: Iespectve of the vales of, the este polyomals sm s ty that s Symmety: () ( ) () Reveso: The polyomal ca be compted by the ecsve elatoshp Devatve: The devatve wth espect to has a ecsve fom () ( ) () () + () d () () () d whee () () 6

7 5--8 () Ed Pots: At, whle all the othe polyomals ae zeo fom the o egatvty ad patto of ty popetes of este polyomals. Ths, P s the ed pot o the eze segmet. Smlaly at, () whle all othe este coeffcets ae zeo, mplyg that Pstheothepotothesegmet. Ed Tagets: The ed tagets have the dectos of P-P ad P-P- espectvely. Geomety Ivaace: De to the patto of ty popety of the este polyomals, the shape of the cve s vaat de otato ad taslato of the coodate fame. Covex Hll Popety: The baycetc(the tem ARYCENTER mples the cete of gavty) ate of este polyomals ese that the eze segmet les wth the covex hll of the cotol pots. Ths popety s sefl tesecto poblems, detecto of tefeece ad povdes estmates of the posto of the cve by comptg the bods of the polygo. Symmety: De to the symmety of este polyomals, f the seqece of the cotol pots s evesed, the symmety of the cve s peseved. P () P ( ) Paamete Tasfomato: At tmes, we may have to expess a eze segmet as a o-omalzed paamete betwee a ad b. Ischacase,set ' a b a No local cotol: The shape of the eze segmet chages globally f aydatapot smoved toaewlocato.fo betweead, evey pot o the old eze segmet gets taslated mplyg that theshapeoftheetecvehaschaged. Vaato Dmshg: Fo a plaa eze segmet, t ca be vefed geometcally that o staght le o that plae tesects wth the segmet moe tmes tha t does wth the coespodg cotol polyle. Smlaly, fo a spatal eze cve, the popety holds fo a plae tesectg the cve ad ts cotol polyle. 7

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