MSG 284 Introduction to Geometric Modelling [Pengenalan kepada Pemodelan Geometri]
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1 UNIVERSITI SAINS MALAYSIA Secod Semester Examato 0/0 Academc Sesso Jue 0 MSG 84 Itrocto to Geometrc Modellg [Pegeala kepada Pemodela Geometr] Durato : hours [Masa : jam] Please check that ths examato paper cossts of EIGHT pages of prted materal before you beg the examato. [Sla pastka bahawa kertas peperksaa megag LAPAN muka surat yag bercetak sebelum ada memulaka peperksaa.] Istructos: [Araha: Aswer all four [4] questos. Jawab semua empat [4] soala.] I the evet of ay dscrepaces, the Eglsh verso shall be used. [Sekraya terdapat sebarag percaggaha pada soala peperksaa, vers Bahasa Iggers hedaklah dgua paka.] /-
2 -- [MSG 84]. (a) Fd a polyomal fucto that terpolates the pots (, ), (, ) ad (4, ). (b) Let P ( u) be a parametrc sple curve defed as P( u) F( u), u a G( u), u a where a. Defe the geometrc cotutes P ( u) at u a. 0 G, G, G of curve (c) Let T, N, B, ad be the ut taget, prcpal ormal, bormal, curvature ad torso of a regular curve P () s at arc legth parameter s 0. Show that the dervatve dn T B. ds [00 marks]. (a) Car fugs polomal yag megterpolas ttk-ttk (, ), (, ) da (4, ). (b) Kataka P ( u) alah satu legkug spl berparameter dtakrf sebaga P( u) F( u), u a G( u), u a d maa a. Takrfka keselajara geometr legkug P ( u) pada u a. 0 G, G, G bag (c) Kataka T, N, B, da alah taget ut, ormal prspal, bormal, kelegkuga da klasa bag suatu legkug alar P () s pada parameter pajag legkok s 0. Tujukka bahawa terbta dn T B. ds [00 markah].../-
3 . (a) Cosder a plae Bézer polyomal of degree P( t) C B ( t), 0 t, 0 -- [MSG 84] where C are cotrol pots ad B () t deote the Berste polyomal ( )! B t t t!( )!, for 0,,,. State the reaso of why the Bézer curve les etrely wth the covex hull of ts cotrol polygo. (b) (c) Gve a cubc Bézer curve 4 P ( t) B0 ( t) B ( t) B ( t) B ( t), 0 t, 4 () where, ad B t, 0,,,, are the Berste polyomals of degree. Calculate the ad such that the curve P () t ca be reced to a quadratc curve. Gve a pecewse curve P( u) F( u), 0 u G( u), u. The curve segmets F ( u) ad G ( u) ca be represeted terms of a local parameter t [0, ] as F( t) C ( t) C t( t) C t, 0 G t t t t t t t t t t ( ) ( ) ( ) ( ) ( ) where C, 0,,. Determe all the C such that P ( u) s a cotuous curve wth the dervatve d P (0) (, 0)., C [00 marks]...4/-
4 -4- [MSG 84]. (a) Pertmbagka satu polomal Bézer satah berdarjah P( t) C B ( t), 0 t, 0 d maa C adalah ttk kawala da B () t meadaka polomal Berste ( )! B t t t!( )!, bag 0,,,. Nyataka sebab megapa legkug Bézer terletak sepeuhya dalam hul cembug polgo kawala. (b) (c) Dber suatu legkug Bézer kubk 4 P ( t) B0 ( t) B ( t) B ( t) B ( t), 0 t, 4 () d maa, da B t, 0,,,, alah polomal Berste berdarjah. Kraka da supaya legkug P () t boleh dkuragka kepada legkug kuadratk. Dber satu legkug bercebsa P( u) F( u), 0 u G( u), u. Legkug segme F ( u) da G ( u) boleh dwakl dalam sebuta parameter setempat t [0, ] sebaga F( t) C ( t) C t( t) C t, 0 G t t t t t t t t t t ( ) ( ) ( ) ( ) ( ) d maa C, 0,,. Tetuka semua C supaya P ( u) alah satu legkug berkeselajara d P (0) (, 0). C dega terbta, [00 markah]...5/-
5 -5- [MSG 84]. Let u (u0, u,, u k) be a o-decreasg kot vector where ad k are postve tegers. The ormalzed B-sple bass fuctos of order k are defed recursvely by ad, u u u N ( u ) 0, otherwse u u u u N ( u) N ( u) N ( u) k k k k u k u u k u where 0,,,., for k, (a) Suppose u (0,, 6, 9). Sketch the fucto fucto values at the gve kots. N 0 () u ad calculate the (b) Cosder a B-sple curve of order o Cartesa plae wth kot vector u (,, 0,,, ) 0 0 P( u) D N ( u) D N ( u) D N ( u), 0 u. () Show that 0 N ( u) N ( u) N ( u ), for 0 u. () Fd the coeffcets D, 0,,, such that the coordate y of P ( u) s o-egatve wth P (0) (, 0), P () (4, ), ad the dot proct of the frst dervatves dp (0) dp () 0 where the legth of vector d P (0) s twce the legth of vector d P (). [00 marks]...6/-
6 -6- [MSG 84]. Kataka u (u0, u,, u k) alah suatu vektor smpula tak meyusut d maa da k adalah teger postf. Fugs asas spl-b terormal berpergkat k dtakrf secara rekurs oleh da, N ( u) 0, u u u d tempat la u u u u N ( u) N ( u) N ( u) k k k k u k u u k u d maa 0,,,., bag k, (a) Adaka u (0,, 6, 9). Lakarka fugs fugs pada smpula yag dberka. N 0 () u da kraka la (b) Pertmbagka suatu legkug spl-b berpergkat pada satah Cartesa dega vektor smpula ( u,, 0,,, ) () 0 0 P( u) D N ( u) D N ( u) D N ( u), 0 u. Tujukka bahawa 0 N ( u) N ( u) N ( u ), utuk 0 u. () Car koefse D, 0,,, supaya koordat y bag P ( u) adalah tak egatf dega P (0) (, 0), P () (4, ), da hasl darab btk bag terbta pertama dp (0) dp () 0 d maa pajag vektor d P (0) adalah a kal pajag vektor d P (). [00 markah]...7/-
7 -7- [MSG 84] 4. (a) Cosder a bquadratc Bézer surface o Cartesa coordate space, j j S( u, v) C B ( v) B ( u), 0 uv,, 0 j 0 where B s () t, 0 t, s 0,,, dcate the Berste polyomals of degree ad C, jare the cotrol pots gve as C 0,0 (,, ), C 0, (,, ), C 0, (,, ), C,0 (,, ), C, (,, 4), C, (,, ), C,0 (4,, ), C, (,, ), C, (4,, ). () Fd the surface pot S ( uv, ) that has maxmum value of coordate z. () Express the curve o surface S alog a parametrc le u v Bézer represetato. (b) Cosder a blearly bleded Coos patch F ( u, v), 0 u, v, whch F (0, 0) (,, ), F (0, ) (, 5, ), F (, 0) (5,, ), F (, ) (5, 5, ). Suppose the boudares of patch F at u 0 ad v 0 are Bézer quadratcs F (0, v) ( v) v( v) 5 v, 5 F ( u, 0) ( u) u( u) u, whle the other two boudares are lear polyomals. Evaluate the ut ormal vector to the patch F at ( u, v ) (0.5, 0.5). [00 marks]...8/-
8 -8- [MSG 84] 4. (a) Pertmbagka suatu permukaa Bézer dwkuadratk pada ruag koordat Cartesa d maa, j j S( u, v) C B ( v) B ( u), 0 uv,, 0 j 0 B s () t, 0 t, s 0 berdarjah da, j,,, meadaka polomal Berste C adalah ttk-ttk kawala yag dberka sebaga C 0,0 (,, ), C 0, (,, ), C 0, (,, ), C,0 (,, ), C, (,, 4), C, (,, ), C,0 (4,, ), C, (,, ), C, (4,, ). () Car ttk permukaa S ( uv, ) yag mempuya la maksmum koordat z. () Nyataka legkug pada permukaa S d sepajag gars parameter u v dalam perwakla Bézer. (b) Pertmbagka satu tampala Coos teraa dwlear F ( u, v), 0 u, v, d maa F (0, 0) (,, ), F (0, ) (, 5, ), F (, 0) (5,, ), F (, ) (5, 5, ). Adaka sempada tampala F pada u 0 da v 0 alah kuadratk Bézer F (0, v) ( v) v( v) 5 v, 5 F ( u, 0) ( u) u( u) u, maakala a sempada la adalah polomal lear. Nlaka vektor ut ormal kepada tampala F pada ( u, v ) (0.5, 0.5). [00 markah] - ooo O ooo -
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