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

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1 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 : jam] Please chec that this examination paper consists of SEVEN pages of printed material before you begin the examination. [Sila pastian bahawa ertas peperisaan ini mengandungi TUJUH mua surat yang berceta sebelum anda memulaan peperisaan ini.] Instructions: Answer all three [] questions. [Arahan: Jawab semua tiga [] soalan.] In the event of any discrepancies, the English version shall be used. [Seiranya terdapat sebarang percanggahan pada soalan peperisaan, versi Bahasa Inggeris hendalah diguna paai]. /-

2 1. (a) State the definition of natural spline function. [MSG 88] (b) Derive the Bézier form of a polynomial function x [ 1, 1]. y( x) (1 x)(1 x ), where (c) Given a rational cubic Bézier curve w C B ( t) wc B ( t) w C B ( t) w C B ( t) R () t, t [, 1] w B ( t) w1 B1 ( t) w B ( t) w B ( t) where Bi () t are the Bernstein polynomials of degree, the coefficients are the Bézier points and the weights w i. Suppose C, C 1, C, C are non-collinear points and w1 w w where w, determine w and w such that R( t) (1 t) C1 tc, for any t (, 1). C i (ii) Derive R ( t) ( x( t), y( t)) which represents a circular arc where x [ 1, 1]. y 1 x, [1 mars].../-

3 [MSG 88] 1. (a) Nyataan tarif fungsi splin asli. (b) Terbitan perwailan Bézier suatu fungsi polinomial dengan x [ 1, 1]. y( x) (1 x)(1 x ), (c) Diberi suatu lengung Bézier ubi nisbah w C B ( t) wc B ( t) w C B ( t) w C B ( t) R () t, t [, 1] w B ( t) w1 B1 ( t) w B ( t) w B ( t) dengan Bi () t polinomial Bernstein berdarjah, peali Bézier dan pemberat w i. C i adalah titi Andaian C, C 1, C, C adalah titi-titi ta olinear dan w1 w w dengan w, tentuan w dan w supaya R( t) (1 t) C1 tc, untu sebarang t (, 1). (ii) Terbitan R ( t) ( x( t), y( t)) yang mewaili lengo bulatan dengan x [ 1, 1]. y 1 x, [1 marah]...4/-

4 4 [MSG 88]. (a) Let P () u, u [, 1], be a cubic B-spline curve defined by a sequence of de Boor control points (1, 1), (1, ), (, ) and (, 1) over a not vector u (,, 1,, 1,,, 4). Evaluate the derivative d P( u) du at u.5. (ii) Suppose a cubic B-spline curve P () u is defined over a not vector u (,, 1,, 1, 1, 1,,, 4), find its de Boor points such that the curve P () u coincides with P () u. (b) Find the de Boor points of a cubic uniform B-spline curve P () u such that the Bézier form of P () u is 1 1 P ( u) (1 u) u(1 u) u, [, 1] u. (c) Figure 1 shows a quadratic B-spline curve defined over the uniformly spaced nots ui i, integers i, and its control polygon D D1 D D D4 D 5. Suppose the point D is duplicated from a single point towards triple points, describe the effect of multiple points D to the B-spline curve at segment u [4, 5]. Setch the resulting B-spline curves and their control polygons when the double D and triple D are used. D 1 D D 5 The given control points. The curve points at nots u i. D D D 4 Figure 1 [1 mars]...5/-

5 5 [MSG 88]. (a) Kataan P () u, u [, 1], ialah suatu lengung splin-b ubi yang ditarif oleh suatu jujuan titi awalan de Boor (1, 1), (1, ), (, ) dan (, 1) pada suatu vetor simpulan u (,, 1,, 1,,, 4). Nilaian terbitan d P( u) du pada u.5. (ii) Andaian suatu lengung splin-b ubi P () u ditarif pada suatu vetor simpulan u (,, 1,, 1, 1, 1,,, 4), cari titi-titi de Boor bagi P () u supaya lengung ini sama dengan P () u. (b) Cari titi-titi de Boor bagi suatu lengung splin-b seragam ubi P () u supaya perwailan Bézier bagi P () u ialah 1 1 P ( u) (1 u) u(1 u) u, [, 1] u. (c) Rajah 1 menunjuan suatu lengung splin-b uadrati yang ditarif pada simpulan seragam ruang ui i, integer i, dan poligon awalannya D D1 D D D4 D 5. Andaian titi D didupliasi daripada titi tunggal e titi ganda tiga, huraian esan titi berganda D terhadap lengung splin-b pada tembereng u [4, 5]. Laar lengung splin-b yang terhasil dan poligon awalannya apabila duaan D dan tigaan D digunaan. D 1 D D 5 Titi awalan Titi lengung pada not u i. D D D 4 Rajah 1 [1 marah]...6/-

6 . (a) A biquadratic Bézier surface is defined by (, ) i, j i ( ) j ( ) 6 [MSG 88] P x y = c B x B y, x 1, y 1 where B s, s, 1,, are the Bernstein polynomials of degree and c, are the Bézier ordinates given as (ii) c, c 1,, c, 1, c, c 1,1 1, c,1, c, c 1,, c, 1., 1,1, 1 Use the de Casteljau algorithm to evaluate the cross boundary derivative to surface P at ( xy, ) (1, 1). Suppose the surface P is represented in parametric B-spline form as ( u, v) i, j Ni ( u) Nj( v) P D, uv, 1 D ij, are the de Boor points and where N s, s, 1,, are the normalized B-spline basis functions of order defined over the nots ui i and vi i, i, 1,,. Determine all the D i, j. i j (b) A quadratic Bézier triangle is defined by where i, j, S( u, v, w ) = C B ( u, v, w), u, v, w 1, u v w 1 i, j, i j i, j, i, j, B,, ( u, v, w ) are the generalised Bernstein polynomials of degree and i j C are the Bézier points given as C,, (, 5, ), 1 C (, 1, ), C,, (1,1, ), 1 1,1,1 C 1,1, (1,, ), 1,,1 C (,1, ), C,, (4,,1). If P (,, z) is a point on the surface S ( u, v, w), evaluate the value z. (ii) Evaluate the directional derivative of S ( u, v, w) with a vector d ( 1, 1, 1) at ( u, v, w ) ( 1,, ). [1 mars]...7/-

7 . (a) Suatu permuaan Bézier biuadrati ditarif oleh (, ) i, j i ( ) j ( ) 7 [MSG 88] P x y = c B x B y, x 1, y 1 dengan B s, s, 1,, polinomial Bernstein berdarjah dan c i, j adalah ordinat Bézier yang diberian sebagai (ii) c, c 1,, c, 1, c, c 1,1 1, c,1, c, c 1,, c, 1., 1,1, 1 Guna algoritma de Casteljau untu menilai terbitan sempadan silang epada permuaan P pada ( xy, ) (1, 1). Andaian permuaan P diwaili dalam bentu splin-b berparameter ( u, v) i, j Ni ( u) Nj( v) P D, uv, 1 D ij, ialah titi de Boor dan dengan N s, s, 1,, adalah fungsi asas splin-b ternormal berperingat yang ditarif pada simpulan ui i dan vi i, i, 1,,. Tentuan semua titi D i, j. (b) Suatu segi tiga Bézier uadrati ditarif oleh dengan i, j, S( u, v, w ) = C B ( u, v, w), u, v, w 1, u v w 1 i, j, i j i, j, i, j, B,, ( u, v, w ) ialah polinomial Bernstein teritla berdarjah dan i j C adalah titi Bézier yang diberian sebagai C,, (, 5, ), 1 C (, 1, ), C,, (1,1, ), 1 1,1,1 C 1,1, (1,, ), 1,,1 C (,1, ), C,, (4,,1). Jia P (,, z) ialah suatu titi pada permuaan S ( u, v, w), nilaian z. (ii) Nilaian terbitan berarah S ( u, v, w) dengan vetor d 1 ( u, v, w ) (,, ). ( 1, 1, 1) pada - ooo O ooo - [1 marah]

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