Geometric Modeling
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1 Geomerc Modelg Crves coed Cc Bezer ad B-Sle Crves Far Chaers Moreso Chaers 4 5
2 4 Tycal Tyes of Paramerc Crves Corol os flece crve shae. Ierolag Crve asses hrogh all corol os. Herme Defed y s edos ad age vecors a edos. Ierolaes all s corol os. o vara der affe rasformaos. Secal case of Bezer ad B-Sle. Bezer* Ierolaes frs ad las corol os. Ivara der affe rasformaos. Crve s age o frs ad las segmes of corol olygo. Easy o sdvde. Crve segme les wh covex hll of corol olygo. Varao-dmshg. Secal case of B-sle. B-Sle* o garaeed o erolae corol os. Ivara der affe rasformaos. Crve segme les wh covex hll of corol olygo. Varao-dmshg. Greaer local corol ha Bezer. *focs of hs lecre sorce: Moreso Agel
3 oao Moreso Corol os ad arameer: ad R Far Corol os ad arameer: ad R sorce: Moreso Far
4 Bezer Crves Far Chaer 4 The de Caselja Algorhm
5 Paraola Cosrco Chage of oao from Chaer revosly: [ [ ow: Add: By sso: sorce: Far
6 The de Caselja Algorhm Gve: se: Examle: cc r r sorce: Far r r r R ad crve o Coceally elega o ecessarly he fases algorhm sorce: Far
7 Some Bezer Crve Proeres Affe varace: Lear erolao s affely vara. Ivarace der affe arameer rasformaos eyod [ o [a: r a Covex hll: Crve says wh covex hll of corol olygo. Each ermedae o s a covex arycerc comao of revosly geeraed os. Plaar corol olygo geeraes laar crve. Helfl for erseco ess. Edo erolao: Crve asses hrogh ad. Verfy sg scheme for = ad =. r a a r sorce: Far
8 Bezer Crves Far Chaer 5- & Moreso Chaer 4 The Berse Form of a Bezer Crve
9 Bezer Geomerc form cc case: B B Berse olyomals. 4 = mer of corol os = degree Evalae a = ad = o show ages relaed o frs ad las corol olygo le segme. sorce: Moreso
10 Bezer Geomerc form geeral case: B B Berse olyomals. + = mer of corol os = degree + Raoal form s vara der ersecve rasformao: where h are rojecve sace coordaes weghs See Chaer of Far for raoal Bezer maeral. h B h B sorce: Moreso
11 More Bezer Crve Proeres Symmery: Berse olyomals are symmerc wh resec o ad -. Covex Hll aga: B B Covex comao so Bezer crve os all le wh covex hll of corol olygo. Bezer crve wh 4 corol os sorce: Far Moreso
12 More Bezer Crve Proeres Degree elevao leavg crve chaged Addg a corol o elevaes degree y. for B ew verces oaed from old olygo y ecewse lear erolao a arameer vales /+. ew corol olygo s covex hll of old oe. sorce: Far
13 More Bezer Crve Proeres Reeaed degree elevao: he lm rodces corol olygo seqece ha coverges o he acal crve. Varao Dmshg: Pecewse lear erolao s varao dmshg. Degree elevao ses ecewse lear erolao. Each sccessve corol olygo cao ersec a gve lae more ofe ha he corol olygo o whch s ased. Ths he crve cao ersec he lae more ofe ha ay of he corol olygos. Corollary: covex corol olygo rodces covex crve segme. sorce: Far
14 Comose Bezer Crves Jog adjace crve segmes s a alerave o degree elevao. Colleary of cc Bezer corol os rodces G coy a jo o: Evalae a = ad = o show ages relaed o frs ad las corol olygo le segme. For G coy a jo o 5 verces ms e colaar. sorce: Moreso
15 B-Sle Crves Moreso Chaer 5 & Far Chaer 8
16 B-Sle Geomerc form o-form o-raoal case where K corols degree K - of ass fcos: k k k f oherwse k k k K are ++K ko vales ha relae o he corol os. Uform case: sace kos a eqal ervals of. Reeaed kos move crve closer o corol os. Cc B-sles ca rovde C coy a crve segme jo os. Raoal form URBS s vara der ersecve rasformao where h are rojecve sace coordaes weghs. See Chaer of Far for raoal B-sle maeral. K Covex comao so B-sle crve os all le wh covex hll of corol olygo. h h K K sorce: Moreso
17 Some B-Sle Crve Proeres Affe varace: Lear erolao s affely vara. Ivarace der affe arameer rasformaos eyod [ o [a: Covex hll: Crve says wh covex hll of corol olygo see laer slde. Each ermedae o s a covex arycerc comao of revosly geeraed os. Plaar corol olygo geeraes laar crve. Helfl for erseco ess. o edo erolao: lke Bezer Varao dmshg Trasformao o some oher aramerc forms: B-Sle crve ca e rasformed o Bezer form. See Moreso. 7. B-Sle crve ca e rasformed o Herme form. See Moreso sorce: Far Moreso
18 Cc B-Sle Geomerc form cc case: 4 sorce: Moreso Evalae a = ad = o show crve edos ad age drecos relaed o corol olygo. 4 B-sle cc ass fcos 4 = mer of corol os = degree + K oherwse f k k k k k k Covex comao so B-sle crve os all le wh covex hll of corol olygo. K 4 Uform cc B-sle:
19 More B-Sle Crve Proeres Symmery: Bass fcos ossess aramerc symmery: Examle: Uform cc B-sle: K K K sorce: Far Moreso
20 Shae Characerzao of Uform Cc B-Sle Crve Segme Shae deeds o corol olygo o-collear corol os deerme ragle Tye of crve segme ad s mooocy deeds o locao of 4 h o relave o ragle : SP: sral corol olygo dces covex moooe crve segme. U: U-shaed covex corol olygo dces covex moooe crve segme. V:V-shaed covex corol olygo dces fleco o a crve segme edo. Crve segme s covex moooe. : -shaed corol olygo dces fleco o wh covex moooe eces o oh sdes of fleco o. SI: self-ersecg corol olygo dces or moooe crve ssegmes. D: self-ersecg corol olygo may dce loo cs or fleco os. I all cases crve segme ca e aroed o a mos moooe eces. SP SP SP SI D SP SP SP U V V sorce: Daels e al. ldg o Wag e al.
21 The deboor Algorhm Gve: kos & corol os & arameer geeralze de Caselja algorhm. Examle: qadrac case wh 4 kos ad corol os sorce: Far [ [ [ [ [ [ crve o Coceally elega o ecessarly he fases algorhm sorce: Far [ [ [ [ [ [ [ [ [ ko seqece = 4 ad =.
22 Geomerc Modelg OeGL Demo o accomay HW#
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