MAT 516 Curve and Surface Methods for CAGD [Kaedah Lengkung dan Permukaan untuk RGBK]

Transcription

1 UNIVERSITI SAINS MALAYSIA Secod Semese Examao / Academc Sesso Jue MAT 56 Cuve ad Suface Mehods fo CAGD [Kaedah Legkug da Pemukaa uuk RGBK] Duao : hous [Masa : am] Please check ha hs examao ae cosss of SEVEN ages of ed maeal befoe you beg he examao [Sla aska bahawa keas eeksaa megadug TUJUH muka sua yag beceak sebelum ada memulaka eeksaa ] Isucos: Aswe all fou [4] quesos [Aaha: Jawab semua ema [4] soala] I he eve of ay dsceaces, he Eglsh veso shall be used [Sekaya edaa sebaag ecaggaha ada soala eeksaa, ves Bahasa Igges hedaklah dgua aka] /-

2 - - [MAT 56] (a) Le ()! Show ha!( )! B () whee B ( ) ( ), Gve B ( ) ( ) () Show ha If, ( ) B ( ) B ( ) B ( ) ( ) B ( ), show ha '() ( )( ) () Idcae wh a dagam he ocess of evaluag he o (5) of a cubc Beze cuve wh cool os,,, usg de Caselau algohm [5 maks] (a) Baka! Tuukka bahawa!( )! () B () dega B ( ) ( ), Dbe B ( ) ( ), () Tuukka bahawa ( ) B ( ) B ( ) B ( ) Jka ( ) B ( ), uukka bahawa '() ( )( ) () Nyaaka dega suau aah, oses mela k (5) suau legkug Beze kubk dega k kawala,,, dega megguaka algoma de Caselau [5 makah] /-

3 - - [MAT 56] (a) Gve a le wh a equao y x ad a aabola ( ) ( ) (,4) ( )(,) (,), () Fd he value of whee he le esecs he aabola ad hece deeme he o of eseco Deeme he o o he aabola such ha he age veco a hs o s omal o he veco defg he sagh le A cubc aoal Tmme cuve wh fou cool os wh wo eal weghs w ad w s gve by ( ) ( ) ( ) 4( ) w 4( ) w ( ), ( ) ( ) 4( ) w 4( ) w ( ) () Gve a examle of he values of he weghs w ad w such ha he o (5) dvdes he segme he ao of : Show ha whe he weghs ae equal ad ad ae aallel he he cubc aoal Tmme cuve gve above wll educe o a coc segme Hece, show ha he weghs w ad w of he coc segme s w w ( )( ) ( )( ) (c) A cubc olyomal f ( u) u u u, u ca be exessed as f ( ) ( u ) y ( ) y ( ) y y, wh ad Fd y, y, y ad [5 maks] (a) Dbe suau gas luus dega esamaa y x da suau aabola ( ) ( ) (,4) ( )(,) (,), () Ca la aabla gas beslag aabola da euka k eslagaya Teuka k ada aabola suaya veco age ada k adalah beseeag dega veco yag meghuaka gas luus 4/-

4 - 4 - [MAT 56] Suau legkug kubk sbah Tmme dega ema k kawala da dua embea dalama w da w dbeka sebaga ( ) ( ) 4( ) w 4( ) w ( ) ( ), ( ) ( ) 4( ) w 4( ) w ( ) () Beka cooh la embea w da w suaya k (5) membahagka segme ada sbah : Tuukka aabla embea dalama adalah sama da da adalah sela maka legkug kubk sbah Tmme yag dbeka daas aka euu keada suau segme ko Juseu, uukka bahawa embea w da w suau segme ko dbe oleh ( )( ) w w ( )( ) (c) Suau olyomal kubk dbeuk oleh f ( u) u u u, u boleh f ( ) ( ) y ( ) y ( ) y y, dega da Ca y, y, y da y [5 makah] u (a) A aoal quadac cuve s defed as () ( ) ( ) wh+ ( ) ( ) w + whee ad w s a osve wegh Le (,), h (,a ), (,) ad he agle h () Deeme he value of he cuvaue () ems of If he cuve eeses a ccula ac, he show ha w cos () If eds o, show ha eds o zeo ad he locao of he o eds o fy 5/-

5 - 5 - [MAT 56] Le wo cuves be gve as ( ) ( ),, ad ( ) ( ) ( ) ( ),, Fd he cuvaue of () a f he os, ad ae collea Hece deeme he locaos of,, ad so ha he cuves () ad () ae oed wh cuvaue couy a (c) A cubc fuco f() o he eval x ca be we as f() ( ) ( ) y ( ) y ( ) y ( ) y ( ) ( ) ( ) ( ) ( ) x whee If he gade of he fuco a x s d we y ems of y, d [5 maks] (a) Suau legkug sbah kuadak dbe sebaga () ( ) ( ) wh+ ( ) ( ) w + dega da w adalah embea osf Adaka (,), h (,a ), (,) da sudu h () Teuka la kelegkuga () dalam sebua Jka legkug mewakl suau legkuk bulaa, maka uukka bahawa w cos () Jka meuu la, uukka bahawa keada la sfa da keduduka k meuu keada f Baka dua legkug dbe sebaga ( ) ( ),, da ( ) ( ) ( ) ( ), Ca la kelegkuga () ada ka, da adalah segas Juseu euka kedudukka,, da suaya legkug () da () adalah ehubug dega keselaaa kelegkuga ada 6/-

6 (c) Suau fugs sbah kubk ada selag x f() [MAT 56] daa duls dalam beuk ( ) ( ) y ( ) y ( ) y ( ) y ( ) ( ) ( ) ( ) ( ) x dega Jka keceua legkug ada x alah d ulska y dalam sebua y, d da [5 makah] 4 (a) Le a suface be defed aamecally as S( u, v) ( u v, u v, u v ), whee uv, () Evaluae he o S (,) Deeme a equao of he age lae a u, v () Exess Su (,) as a quadac Beze cuve wh suable cool os A B-sle quadac cuve wh ko veco {,,,,,,}ad cool os {,,, } s gve by Whee Nd, ( ) N ( ), N (), f [ ), elsewhee N () N N d, d, d, d d d () Evaluae N, ad N 4, Exess () ems of he cool os,, wh ad d 4 (a) Adaka suau amala emukaa dakf seca beaamee S( u, v) ( u v, u v, u v ), dega uv, () Nlaka k S (,) Teuka esamaa saah age ada u, v () Ugkaka Su (,) sebaga suau legkug kuadak dega meyaaka k kawala yag besesuaa [5 maks] 7/-

7 - 7 - [MAT 56] Suau sl B kuadak dega veco ko {,,,,,,} da k kawala {,,, } adalah dbe sebaga dega N (), ( ) N ( ),, f [ ) la ema N () N N d, d, d, d d d d () Nlaka N, da N 4, dega da Ugkaka d () alam sebua k kawala,, [5 makah] - ooo O ooo -