MSG 388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik untuk Grafik Komputer]

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Leon Mosley
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1 UNIVERSITI SAINS MALAYSIA Frst Semester Examnaton 0/0 Academc Sesson January 0 MSG 88 Mathematcal Algorthms for Computer Graphcs [Algortma Matemat untu Graf Komputer] Duraton : hours [Masa : jam] Please chec that ths examnaton paper conssts of SEVEN pages of prnted materal before you begn the examnaton. [Sla pastan bahawa ertas pepersaan n mengandung TUJUH mua surat yang berceta sebelum anda memulaan pepersaan n.] Instructons: [Arahan: Answer all three [] questons. Jawab semua tga [] soalan.] In the event of any dscrepances, the Englsh verson shall be used. [Seranya terdapat sebarang percanggahan pada soalan pepersaan, vers Bahasa Inggers hendalah dguna paa]. /-

2 -- [MSG 88]. (a) Fnd a Bézer functon that concdes wth a polynomal y ( x )( x ), for x. (b) Fgure shows the control polygon of a quadratc Bézer curve P defned wth control ponts C 0, C and C as 0 P( t) ( t) C t( t) C t C, t [0, ]. Suppose the ponts K and L are defned by K ( t) C tc, 0 L ( t) C tc. Derve the frst order dervatve of P n terms of K and L. C K L C 0 C Fgure (c) Gven a quadratc ratonal Bézer curve xt () C B ( t) wc B ( t) C B ( t) R () t, t [0, ] yt () B ( t) wb ( t) B ( t) where w 0, C 0 (, ), C (, ), C (, ) and! B ( t) ( t) t!( )!, 0,,. () Evaluate the weght w such that the curve R touches the x -axs at a pont. () Suppose the pont C s shfted to poston (, ), evaluate the w such that the curve R s a crcular arc. [00 mars].../-

3 -- [MSG 88]. (a) Car fung Bézer yang sama dengan polnomal y ( x )( x ), bag x. (b) Rajah menunjuan polgon awalan bag lengung Bézer uadrat P yang dtarf dengan tt awalan C 0, C dan C sebaga 0 P( t) ( t) C t( t) C t C, t [0, ]. Andaan tt K dan L dtarf oleh K ( t) C tc, 0 L ( t) C tc. Terbtan terbtan pertama P dalam sebutan K dan L. C K L C 0 C Rajah (c) Dber suatu lengung Bézer nsbah uadrat xt () C B ( t) wc B ( t) C B ( t) R () t, t [0, ] yt () B ( t) wb ( t) B ( t) d mana w 0, C 0 (, ), C (, ), C (, ) dan! B ( t) ( t) t!( )!, 0,,. () Nlaan pemberat w supaya lengung R menyentuh pas- x pada satu tt. () Andaan tt C dalh e eduduan (, ), nlaan w supaya lengung R alah satu lengo bulatan. [00 marah].../-

4 -- [MSG 88]. (a) Fgure shows a par of trangles wth vertces A (, ), B (, 0), C (, ) and D (, ). Let P be the centrod of the trangle BCD. Fnd the barycentrc coordnates of pont P wth respect to the trangle ABC. C A P D B Fgure (b) Gven a cubc Bézer trangular patch, j, 0 j, j,, j, S( u, v, w ) = C B ( u, v, w), 0 u, v, w, u v w where, j, C and! j B, j, ( u, v, w ) = u v w.! j!! Let C,0,0 (,, ), C,,0 (,, ), C,0, (,, ), C,, (,, ), C 0,,0 (,, ) and C 0,0, (,, ). () Determne the coeffcents C,,0, C,0,, C 0,, and C 0,, such that the patch S can be reduced to a quadratc Bézer patch. () Suppose C,,0 (,, ), C,0, (,, ), C 0,, (,, ) and C 0,, (,, ). Use the de Casteljau algorthm to evaluate the surface pont S at (,, ) (/, / 6, / ) u v w. [00 mars]...5/-

5 -5- [MSG 88]. (a) Rajah menunjuan sepasang seg tga dengan bucu A (, ), B (, 0), C (, ) dan D (, ). Kataan P alah sentrod seg tga BCD. Car oordnat barpusat tt P terhadap seg tga ABC. C A P D B Rajah (b) Dber suatu tampalan seg tga Bézer ub, j, 0 j, j,, j, S( u, v, w ) = C B ( u, v, w), 0 u, v, w, u v w d mana C dan, j,! j B, j, ( u, v, w ) = u v w.! j!! Kataan C,0,0 (,, ), C,,0 (,, ), C,0, (,, ), C,, (,, ), C 0,,0 (,, ) dan C 0,0, (,, ). () Tentuan oefsen C,,0, C,0,, C 0,, dan C 0,, supaya tampalan S boleh durangan epada tampalan Bézer uadrat. () Andaan C,,0 (,, ), C,0, (,, ), C 0,, (,, ) dan C 0,, (,, ). Gunaan algortma de Casteljau untu menla tt permuaan S pada ( u, v, w ) (/, / 6, / ). [00 marah]...6/-

6 -6- [MSG 88]. (a) Gven a polygon of 0 ponts n Fgure. Fnd the number of ponts produced after the polygon s refned twce by Chan s subdvson scheme. Fgure (b) Gven a cubc B-splne curve defned on a non-decreasng not vector u (u0, u,, u 7 ) as P( u) D N ( u), u u u 0 where D are dstnct de Boor ponts. The functons formulated recursvely by and () u u u u N ( u) N ( u) N ( u) u u u u, u u u N ( u ) 0, otherwse.,, N () u can be Descrbe the condtons on the not vector u such that the curve P nterpolates the ponts D 0 and D. () Suppose the curve P s defned by D 0 (, ), D (, ), D (, ) and D (, ) over u (,,, 0,,,, ). A set of new de Boor ponts s ganed when a not value u 0.75 s nserted twce nto u wthout changng the shape of P. Fnd the postons of these new de Boor ponts. () Suppose u (,, 0.5, 0,,,, 5), fnd the de Boor ponts D, 0,,,, such that the curve 6 9 P ( u) u u u u u u. [00 mars]...7/-

7 -7- [MSG 88]. (a) Dber suatu polgon 0 tt dalam Rajah. Car blangan tt yang dhaslan selepas polgon dperhalus dua al oleh sema subdvs Chan. Rajah (b) Dber suatu lengung spln-b ub yang dtarf pada vetor smpulan ta menyusut u (u0, u,, u 7 ) sebaga 0 P( u) D N ( u), u u u d mana D adalah tt-tt de Boor yang berbeza. Fungs drumusan secara reurs oleh dan u u u u N ( u) N ( u) N ( u) u u u u, N ( u) 0, u u u d tempat lan.,, N () u boleh () Nyataan syarat pada vetor smpulan u supaya lengung P mengnterpolas tt D 0 dan tt D. () Andaan lengung P dtarf dengan D 0 (, ), D (, ), D (, ) dan D (, ) pada u (,,, 0,,,, ). Satu set tt de Boor baru dperoleh apabla nla smpulan u 0.75 dmasuan dua al e dalam u tanpa mengubah bentu P. Car eduduan tt de Boor baru n. () Andaan u (,, 0.5, 0,,,, 5), car tt-tt de Boor D, 0,,,, supaya lengung 6 9 P ( u) u u u u u u. - ooo O ooo - [00 marah]

MSG 388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik untuk Grafik Komputer]

MSG 388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik untuk Grafik Komputer] UNIVERSITI SAINS MALAYSIA First Semester Examination 9/1 Academic Session November 9 MSG 88 Mathematical Algorithms for Computer Graphics [Algoritma Matemati untu Grafi Komputer] Duration : hours [Masa