MSG 284 Introduction to Geometric Modelling [Pengenalan kepada Pemodelan Geometri]
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1 UNIVERSITI SAINS MALAYSIA Secod Semester Examato / Academc Sesso Jue MSG 84 Itroducto to Geometrc Modellg [Pegeala kepada Pemodela Geometr] Durato : hours [Masa : jam] Please check that ths examato paper cossts of SEVEN pages of prted materal before you beg the examato. [Sla pastka bahawa kertas peperksaa megadug TUJUH muka surat yag bercetak sebelum ada memulaka peperksaa.] Istructos: [Araha: Aswer all three [] questos. Jawab semua tga [] 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) Let P () s be a regular curve terms of the arc legth parameter s. Show that the ut taget T, the prcpal ormal N ad the bormal B of curve P are mutually orthogoal. (b) Gve a crcle x y x, rewrte the crcle stadard form ad the evaluate ts curvature. (c) Fd a polyomal fucto that terpolates the pots (, ), (, ) ad (4,). (d) Let P ( u) be a pecewse curve composed of quadratc polyomal F ( u) ad cubc polyomal G ( u) as P( u) x( u), y( u) F( u), u G( u), u o the xy -plae. Wth x( u) u, determe the F ad G terms of u such that the curve P s ad P () (, ), P () (, ), P () (4,) d () (, ) du P. G geometrcally cotuous at u ad satsfes [ marks].../-
3 -- [MSG 84]. (a) Kataka P () s alah satu legkug alar berparamater pajag legkok s. Tujukka bahawa tage ut T, ormal prspal N da bormal B bag legkug P adalah salg berkeserejaga. (b) Dber bulata x y x, tuls semula bulata dalam betuk pawa kemuda laka kelegkugaya. (c) Car fugs polomal yag megterpolas ttk-ttk (, ), (, ) da (4,). (d) Kataka P ( u) alah satu legkug bercebsa yag dgubah dega polomal kuadratk F ( u) da polomal kubk G ( u) sebaga P( u) x( u), y( u) F( u), u G( u), u pada satah- xy. Dega x( u) u, tetuka F da G dalam sebuta u supaya legkug P adalah selajar secara bergeometrc memeuh P () (, ), da P () (, ), P () (4,) d () (, ) du P. G pada u da [ markah]...4/-
4 . (a) Let the Berste polyomal of degree be deoted as Gve ( )! B t t t!( )! B ( t ), show that -4- [MSG 84], t, for,,,. B () t t. (b) Fd a cubc Bézer fucto that matches wth y( x) x, x. (c) Gve a quadratc Bézer curve P ( t) B ( t) B ( t) B ( t), t, where B () t,,,, are the Berste polyomals of degree. Suppose the curve s trucated to a small curve segmet the parametrc terval.5 t.75, fd the three ew Bézer pots defg that curve segmet. (d) Gve a fuctoal bquadratc Bézer surface z( x, y) C B ( y) B ( x ), x, y, where j, j j B s () t, t, s,,, are the Berste polyomals of degree C are the Bézer ordates gve as ad, j C,, C,, C,, C,, C,, C,, C,, C,, C,. Determe the ut ormal vector of taget plae to the surface z at ( x, y ) (.5,.5). [ marks]...5/-
5 -5- [MSG 84]. (a) Kataka polomal Berste berdarjah dtada sebaga Dber ( )! B t t t!( )! B ( t ), tujukka bahawa, t, utuk,,,. B () t t. (b) Car satu fugs Bézer kubk yag memadaka y( x) x, x. (c) Dber satu legkug Bézer kuadratk P ( t) B ( t) B ( t) B ( t), t, d maa B () t,,,, adalah polomal Berste berdarjah. Adaka legkug dsgkatka kepada segme legkug kecl dalam selag.5 t.75, car tga ttk Bézer baru yag meakrf segme legkug berkeaa. (d) Dber satu permukaa Bézer dwkuadratk fugsa z( x, y) C B ( y) B ( x ), x, y, d maa j, j j B s () t, t, s,,, adalah polomal Berste berdarjah C alah ordat Bézer dberka sebaga da, j C,, C,, C,, C,, C,, C,, C,, C,, C,. Tetuka vektor ut ormal bag satah tage kepada permukaa z pada ( x, y ) (.5,.5). [ markah]...6/-
6 -6- [MSG 84]. (a) Let u (u, 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 ), otherwse u u u u N ( u) N ( u) N ( u) k k k k u k u u k u where,,,. Gve a B-sple curve of order, for k, P ( u) N ( u) N ( u) N ( u) N ( u), u u u 4. () Suppose u (,,,, 6), fd the local support of fucto Next, show that P ( u) s a C cotuous curve. () Suppose u (,,,,,, 4), fd the curve pot P at u. N () u. (b) Cosder a blearly bleded Coos patch F ( u, v), u, v, whch F (, ) (,, ), F (, ) (, 5, ), F (, ) (5,, ), F (, ) (5, 5, ). Suppose the boudary of patch F at u s a Bézer quadratc F (, v) ( v) v( v) 5 v, whle the other three boudares are lear polyomals. Evaluate the pot o patch F at ( u, v ) (.5,.5). [ marks]...7/-
7 -7- [MSG 84]. (a) Kataka u (u, u,, u k) alah suatu vektor kot tak meyusut d maa da k adalah ombor teger postf. Fugs asas spl-b terormal berpergkat k dtakrf secara rekurs sebaga da N () u =,, 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, utuk k, d maa,,,. Dber satu legkug spl-b berpergkat P ( u) N ( u) N ( u) N ( u) N ( u), u u u 4. () Adaka u (,,,, 6), car sokoga setempat bag fugs N () u. Seterusya, tujukka bahawa P ( u) alah satu legkug berkeselajara C. () Adaka u (,,,,,, 4), car ttk legkug P pada u. (b) Pertmbagka satu tampala Coos teradua dwlear F ( u, v), u, v, d maa F (, ) (,, ), F (, ) (, 5, ), F (, ) (5,, ), F (, ) (5, 5, ). Adaka sempada tampala F pada u alah satu kuadratk Bézer F (, v) ( v) v( v) 5 v, maakala tga sempada la adalah polomal lear. Nlaka ttk tampala F pada ( u, v ) (.5,.5). [ markah] - ooo O ooo -
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