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1 Compue Gaphcs Geomec Moelg Iouco - Geomec Moelg (GM) sce e of 96 - Compue asssace fo - Desg: CAD - Maufacug: : CAM - Moels: - Classcal: : Masemoel (cla( cla, poopes,, Mock-up) - GM: mahemacal escpo fo oec epeseao - pcal applcao aeas: - Ca- a plae esg, egeeg - Eeame Iouco Geomec Moelg Cuves - Cuve s a se of pos - Ofe gve paamec epeseao { p h( [ a, ]}, h : R R Moelg of cuves a sufaces - Compue Ae Geomec Desg - aamec escpo Sol moelg - Cosucve Sol Geome (CSG) - Logcal opeaos o ase sols - Eample: ( f ( s(, ( f ( cos(, [,π ] - Iepee vaale - hscal moel: : me, moveme of pacle olomal cuves, moom ass - aamec cuves ese poloms a aoal fucos - Commol efe ove segmes - Low egee - oloms of egee : - Effce evaluao (oe-scheme): c (...(( c + c ) + c ) c) + c olomal cuves, moom ass - Moom-ass fo : {{ } - Smple poloms: - Lea: ) c + c - Quaac: ) c + c + c - Cuc: ) c + c + c + c (alo-ass) Le hough pos aaola hough pos hough 4 pos
2 Iepolao - Movao: esmae values ewee kow values (cool pos) - Ofe polomal epolao - Iepolao of fuco values: - Gve: + pas (, f ) - Seach: : olom ) of f egee wh ) f f fo all cool pos Iepolao - Cuve epolao: - Gve: + pos a - Seach: polomal cuve of egee : ) a - Ko veco (,,,... ) - Repaameao esuls ffee cuve a a f 4 - hs polom s uquel efe a a Moom epolao - Appoach: Moom-ass: {{ } c Cool pos ass - Fom ) a he ssem of equaos s eve: a a M a L c compoes pe e M M M O M c M M M O M M c Vaemo Ma Moom epolao - Dsavaages: - Solvg he ssem of equaos s umecall comple - No affe vaa - No uve cool, geomec epeao of cool pos - Sum of pos u o vecos - No local cool - aso ewee segmes ol C -couous.g. - Oscllaos f ( ) + [ 5,5] Moom epolao - Eample: cuc epolao polom - Cuves of lowes egee,, wch ae o plaa D ( a + + c + ( a + + c + ( a + + c + - Ca e we ma fom: Q () [ (), (), () ] C a C c a c [,,], a c Moom epolao - C ca e spl o M (ass ma) G (geome veco) a C c - M s fe fo a gve epolao scheme - G epes o he epolao cosas Eample: : lea epolao a c a M G c
3 Moom epolao - Lea epolao (gve epos a ) Q( a a [ ] [ ] M G eme epolao - Goal: cuc polomal cuve efe pos ) a ) wh evaves () ( ) - Appoach: () () ( + + ) ) - Solve fo a so coeffces els: c ( wh coeffces c ), c (), c (), c ), eme epolao () () G R () R R () R,, R, R ), ), () () Q() () () () R R [,,, ] M G [,,, ] M G [,,, ] M G [,,, ] M G G M G [,, ] [ ],,, M G eme epolao R R G M G Q( R R R M ( [ + ) + ( ] + [ ) ] + ( + ) R + ( Q( (, (, (,,, ) eme epolao - Cuc eme-oloms oloms: ( ( (+ ) ( ( ( ( ( ( ) eme epolao - Eample: Chales eme (8-9)
4 eme epolao - Coveso ewee Moom- a eme-ass ass: ( ) M GM ass veco Geome veco eme epolao - Repeseao of eme-ass Moom-ass ass: + ( ) + M M M + M M - Coveso o Moom ass ma fom: Moomepeseao emeepeseao ( ( ( ( ) ( )) () () p () G - Coveso ma s possle geeal also fo ohe pes of cuves M M M G eme epolao ée-cuves - opees: - Nehe affe vaa wh espec o cool pos o wh espec o vecos - No local cool - Dffcul o f age vecos - Cuve segmes ca e aache couousl - Iepolao ewee pos wh ages, e.g.. fo Kefame-Amao wh gve poso a veloc - Iea: age vecos efe fs a las wo pos: - a wll e epolae - a wll e appomae - Relao o eme-iepolao Iepolao: ) ) () ( ) () ( ) wh (egee) cuc ée-cuve ée-cuves ée-cuves - a eme-iepolao ) () G () p () - A fo he cuve: M G M M M M M M M M M G G M wh M M G Geome veco fo ée Ma fo ée o eme 6 - he cuc ée-cuve: ( + ( + ( + ( - ese-oloms of egee : wh oma [,] ese-oloms ( ( ) wh ée-cool-os wh!!( )! 4
5 ée-cuves - Cuc ese-oloms: ( ( ( ( ( ( ( Mamum of ( s a ée-cuves - Of egee : lea epolao p ( Of egee : ( ) ( ) ( ) ( ) (, ( (, ( ( + ( + ( + ( ( + ( + ( + ( (( + ) + (( + ) eae lea epolao ée-cuves ée-cuves - Ieae lea epolao fo egee : p ( ( + Eample fo,4 - opees: - ao of u (affe vaa - osveess ( ( - Smme ( ( - Recuso ( ( ( + ( wh (, a ( fo {,, K, } - ass fo veco space of polomals of egee ée-cuves - e Caselau - Iea: ecuso/eave lea epolao - Gve he ée-os (cool pos) ée-cuves - e Caselau - Apple fo ee-cuves a e Caselau algohm: { E {,, K, }) - Se he e Caselau-os os: ( ( ( ( + (, + {, K, }, {, K, } - he, ( ) esces he po o he cuve ha coespos o paamee value 5
6 ée-cuves - e Caselau - Scheme fo e Caselau fo egee : ée-os e Caselau-os ée-cuves - e Caselau - Relao o ese-oloms: ( + (, {, K, }, {, K, } - he ée-cuve hs oao: - os - ée pos - : ( ( ( ( e-cuve ca pos of oe o (, ) also epese ée-cuves ca e cosuce fom e Caselau o ese-oloms of egee ( ( ( ée-cuves - Affe Ivaace: - Affe mappgs echage wh cuve cosuco: Φ( ( ) Φ( ) ( - Reaso: pao of u ue o ese-oloms - Reaso (aleav): e Caselau s affe vaa ée-cuves - Cove-hull pope - Cuve emas cove hull of cool polgo - Reaso: : ese-oloms ae posve - Reaso (aleav): e Caselau geeaes cove comaos ol - Smplfes eseco es e.g. wh as (smple fs es wh oug o of cool polgos) ée-cuves ée-cuves - Vaao Dmshg ope: - Aa plae R esecs cuve o moe ha esecs he cool polgo - Lea pecso: - Equsa ée-os o a le el (ec paamee) le hough hese pos - No local cool pseuo-local cool: - Chage of a ée-os chages ee cuve - Doma couo aou mamum / of coespog ese-olom - Iepolao: a ae pos o he cuve - age pope & ( ) ( ), & () ( ) - ages o cuve: - Devaves of ese-oloms: ( ( ( ( ) - Devave of cuve: & ( ( Δ ( ( + ) m Δ ( + ( ) ( Δ ) age veco fo cuve of egee - 6
7 ée-cuves - Eamples: ée-cuves (Suvso) - Wh e Caselau a ée-cuve ca e spl o wo segmes, whch ae aga ée-fom - New ée-os of segmes coss of uppe a lowe oua of e Caselau-Scheme suvso of cuve ée-cuves (Suvso) - Refe cool polgos covege agas ée- Cuve - accal ssues: - Spl paamee eval o equal halfs - Epoeal covegece - Effce scheme o epese cuve meas of pecewse lea epolao - Effce scheme fo eseco es oug o of efe segmes - Desg-efeme local cease of ume of cool pos ée-cuves os & Cos - os: - ée-olgo gves ovevew of appomae shape of cuve - Iuve a coollale chage of cuve - Smple a effce algohms fo epeseao a eseco of cuve - Devave ca e compue - Cos: - Degee of cuve s couple o ume of cool pos,.e.. hgh polomal egee - No local cool: chage of oe po chages ee cuve Sple-Cuves - olem so fa: polom egee epes o ume of cool pos - Iea: - Mulple segmes wh low egee sea of oe segme of hgh egee - Impoa s smooh aso ewee segmes - Sple: - A h flele o use fo he cosuco of shps - Deusch: Saklae, Sakfukoe Sple-Cuves - Aoal movao: - A cuc Sple esces he shape of a h o ha s fe a sa a e po - Mmes eg eeg l E c κ ( s) s (κ s cuvaue of cuve) - Segmes ca e of aa pe: - eme-cuves - Quacs - ée-cuves 7
8 ée-sples -Sples - Sple-Segmes Segmes : s ( ( u) - Sple s(u) ) s sum of segmes,,,, u u +, +, - Eample: C -couous cuve fom pecewse quaac ée-segmes - Ie pos ae eua - Cool pos,,, 5, 7, 8 esce he cuve eel , Sufaces esopouc paches - ecewse polomals f: R R - Geeale ee cuves - esopouc paches - agle paches (o scusse hee) - Cuve-valuevalue cuves - ee cuve: ( - Cool pos move o cuve u, v) ( u) ( v) ( v) ( u) m m esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus 8
9 esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus 9
10 esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus esopouc paches esopouc paches - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - Caselau Algohm - u-v v oe: ++ uvaae Caselaus - v-u u oe: m++ uvaae Caselaus esopouc paches esopouc paches - De Caselau algohm - Dec lea - De Caselau algohm - Dec lea
11 esopouc paches esopouc paches - De Caselau algohm - Dec lea - De Caselau algohm - Dec lea esopouc paches - De Caselau algohm - Dec lea
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