UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 015/016 Academic Sessio Jue 016 MSG388 Mathematical Algorithms for Computer Graphics [Algoritma Matematik utuk Grafik Komputer] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists of SEVEN pages of prited material before you begi the examiatio [Sila pastika bahawa kertas peperiksaa ii megadugi TUJUH muka surat yag bercetak sebelum ada memulaka peperiksaa ii] Istructios: Aswer FIVE (5) questios [Araha: Jawab LIMA (5) soala] I the evet of ay discrepacies, the Eglish versio shall be used [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai] /-
- - 1 (a) Figure 1 shows the cotrol polygo of a Bézier curve P () t, t [0, 1] Use the de Casteljau s Algorithm to calculate the poit o the curve P at t 075 (1, ) (3, ) (1, 1) (, 1) Figure 1 (3, 1) Give a splie fuctio 3 x x x x 4 1 1 8, 1 3 3 b0 (1 x) 3 b1 x(1 x) 3 b x (1 x) b3x, 1 x 0 y 3 3 c0 (1 x) 3 c1x(1 x) 3 cx (1 x) c3x, 0 x 1 3 x 6x 9x 8, 1 x 3 Determie all the coefficiets b i ad c i, i 0, 1,, 3, such that the splie fuctio is C cotiuous o (, 3) 1 (a) Rajah 1 meujukka poligo kawala suatu legkug Bézier P () t, t [0, 1] Guaka Algoritma de Casteljau utuk megira titik legkug P pada t 075 (1, ) (3, ) (1, 1) (, 1) (3, 1) Rajah 1 3/-
- 3 - Diberi suatu fugsi spli 3 x x x x 4 1 1 8, 1 3 3 b0 (1 x) 3 b1 x(1 x) 3 b x (1 x) b3x, 1 x 0 y 3 3 c0 (1 x) 3 c1x(1 x) 3 cx (1 x) c3x, 0 x 1 3 x 6x 9x 8, 1 x 3 Tetuka semua pekali b i da c i, i 0, 1,, 3, supaya fugsi spli adalah berkeselajara C pada (, 3) Let V 1 (0, 3), V (, 0) ad V 3 (, 0) be the vertices of a triagle T i two dimesio For ay poit P ( x, y) i T, P uv vv wv 1 3 where ( u, v, w ) are the barycetric coordiates of P with respect to the vertices of T A surface S( u, v, w) 4u luv 4v lvw 4w l wu is defied o T where l (a) If l, evaluate S at ( x, y) (1, 1) Fid the value of l if the miimum value S is 1 4/-
- 4 - Kataka V 1 (0, 3), V (, 0) da V 3 (, 0) ialah bucu-bucu segi tiga T dalam dua dimesi Sebarag titik P ( x, y) dalam T, P uv vv wv 1 3 di maa ( u, v, w ) adalah koordiat baripusat bagi P terhadap bucu-bucu T Permukaa S( u, v, w) 4u luv 4v lvw 4w l wu ditakrif pada T di maa l (a) Jika l, ilaika S pada ( x, y) (1, 1) Cari ilai l jika ilai miimum S ialah 1 3 Let the Berstei polyomials of degree be deoted by ( )! i Bi t t 1 t i!( i)! i, 0t 1, for i 0, 1, K, (a) Figure shows a ratioal quadratic 0 0 1 1 B0 ( t) B1 ( t) B ( t) C B ( t) C B ( t) C B ( t) R () t, t [0, 1] ad its cotrol polygo M be the midpoit of the lie segmet C 0 to C Suppose the ratioal curve divides the lie segmet C 1 to M with ratio :3, fid the value of C 1 R C 0 M C Figure 5/-
- 5 - Give a ratioal quadratic 1 t rt (), [0, 1] t 3t 1 t Fid a ratioal cubic Bézier which is idetical to r 3 Kataka poliomial Berstei berdarjah ditadaka sebagai ( )! i Bi t t 1 t i!( i)! i, 0t 1, bagi i 0, 1, K, (a) Rajah meujukka satu kuadratik isbah 0 0 1 1 B0 ( t) B1 ( t) B ( t) C B ( t) C B ( t) C B ( t) R () t, t [0, 1] da poligo kawalaya M ialah titik tegah segme garis C 0 ke C Adaika legkug isbah membahagika segme garis C 1 ke M dega isbah :3, cari ilai C 1 R C 0 M Rajah C Diberi satu kuadratik isbah 1 t rt (), [0, 1] t 3t 1 t Cari satu Bézier kubik isbah yag serupa dega r 6/-
- 6-4 The ormalised basis splie of order k o a o-decreasig kot vector u (u0, u,, u ), m k, ca be formulated recursively by 1 ad m u u u u N N N k i k1 ik k1 i uik 1ui i uik ui1 i 1 1, 1 ui u u Ni 0, otherwise Give a cubic B-splie curve i1 4 4 1 4 1 4 4 4 P N0 N1 N N3 1 4 1 4 with the kot vector u (, 1, 0, 0, 1, 3, 4, 6), for k 1, (a) Use the de Boor Algorithm to calculate the poit P at u 05 Fid a cubic Bézier curve which is idetical to P 4 Spli asas terormal berperigkat k pada suatu vektor simpula tak meyusut u (u 0, u 1,, u m), m k, boleh dirumuska secara rekursif oleh da u u u u N N N k i k1 ik k1 i uik 1ui i uik ui1 i 1 1, 1 ui u u Ni 0, sebalikya Diberi satu legkug spli-b i1 4 4 1 4 1 4 4 4 P N0 N1 N N3 1 4 1 4 dega vektor simpula u (, 1, 0, 0, 1, 3, 4, 6), bagi k 1, (a) Guaka Algoritma de Boor utuk megira titik P pada u 05 Cari satu legkug Bézier kubik yag serupa dega P [100 markah] 7/-
- 7-5 (a) A uiform B-splie curve of order 3 is defied with the cotrol polygo show i Figure 3 Sketch the curve ad specify the edpoits of each curve segmet Use Chaiki Subdivisio Algorithm to obtai a set of ew vertices from the give cotrol polygo i Figure 3 Sketch the polylie coectig the ew vertices D D D 0 5 D 3 D 1 Figure 3 D 4 5 (a) Satu legkug spli-b seragam berperigkat 3 ditakrif dega poligo kawala yag ditujukka dalam Rajah 3 Lakarka legkug itu da yataka titik-titk hujug bagi setiap segme legkug Guaka Algoritma Subdivisi Chaiki utuk medapatka satu set bucubucu baru dari poligo kawala yag diberika dalam Rajah 3 Lakarka poligaris yag meghubugka bucu-bucu baru D D D 0 5 D 3 D 1 Rajah 3 D 4 - ooo 0 ooo -