Matrices in Computer Graphics
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1 Marce n Compuer Graphc Tng Yp Mah 8A //
2 Tng Yp Mah 8A Abrac In h paper, we cu an eplore he bac mar operaon uch a ranlaon, roaon, calng an we wll en he cuon wh parallel an perpecve vew. Thee concep commonl appear n veo game graphc. Inroucon The ue of marce n compuer graphc weprea. Man nure le archecure, caroon, auomove ha were formerl one b han rawng now are one rounel wh he a of compuer graphc. Veo gamng nur, mabe he earle nur o rel heavl on compuer graphc, now repreenng renere polgon n - Dmenon. In veo gamng nur, marce are maor mahemac ool o conruc an manpulae a realc anmaon of a polgonal fgure. Eample of mar operaon nclue ranlaon, roaon, an calng. Oher mar ranformaon concep le fel of vew, renerng, color ranformaon an proecon. Uneranng of marce a bac nece o program D veo game. Graphc Screenho aen from Operaon Flahpon Polgon fgure le hee ue man fla or conc urface o repreen a realc human oler. The la coornae a calar erm. Homogeneou Coornae Tranformaon Pon,, n R can be enfe a a homogeneou vecor,,, h,,, wh h h h h on he plane n R 4. If we conver a D pon o a 4D vecor, we can repreen a ranformaon o h pon wh a 4 4 mar.
3 Tng Yp Mah 8A Tranformaon of Pon In general, ranformaon of pon can be repreene b h equaon: Tranforme Pon Tranformaon Mar Orgnal Pon In a more eplc cae, a plane panne b wo vecor can be repreene b h equaon: f c e b a Mar on Tranforma f e, c b a pan Mar on Tranforma Plane Orgnal Mar on Tranforma Plane Tranforme Repreenaon of a plane ug marce EXAMPLE Pon,, 6 n R a Vecor,, 6, or 4,,, n R 4 NOTE I poble o appl ranformaon o D pon whou converng hem o 4D vecor. The raeoff ha ranformaon can be one wh a gle mar mulplcaon afer he converon of pon o vecor. More on h afer Tranlaon. an are calar c b a f e c b a f e
4 Tng Yp Mah 8A 4 Tranlaon A ranlaon bacall mean ang a vecor o a pon, mang a pon ranform o a new pon. Th operaon can be mplfe a a ranlaon n homogeneou coornae,,, o,,,. Th ranformaon can be compue ug a gle mar mulplcaon. Tranlaon Mar for Homogeneou Coornae n R 4 gven b h mar:,, T Gven an pon,, n R, he followng wll gve he ranlae pon. For a phere o move o a new poon, we can hn of h a all he pon on he phere move o he ranlae phere b ang he blue vecor o each pon.
5 Tng Yp Mah 8A Graphc Screenho aen from Operaon Flahpon In veo game, obec le arplane ha oen change hape namcall rg bo ue Tranlaon o move acro he. All he pon ha mae up he plane have o be ranlae b he ame vecor or he mage of he plane wll appear o be reche. NOTE If we have more han one pon, we woul have o appl h aon o ever pon. & a a Wh homogeneou coornae, we can ue a gle mar mulplcaon. A we can ee, lnear em eaer o olve wh homogenenou coornae ranformaon.
6 Tng Yp Mah 8A 6 Scalng Scalng of an menon requre one of he agonal value of he ranformaon mar o equal o a value oher han one. Th operaon can be vewe a a calng n homogeneou coornae,,, o,,,. Value for,, greaer han one wll enlarge he obec, value beween ero an one wll hrn he obec, an negave value wll roae he obec an change he e of he obec. Scalng Mar for Homogeneou Coornae n R 4 gven b h mar:,, S Gven an pon,, n R, he followng wll gve he cale pon. If we wan o cale he heaheron proporonall, we appl he ame calng mar o each pon ha mae up he heaheron.
7 Tng Yp Mah 8A Roaon Roaon are efne wh repec o an a. In menon, he a of roaon nee o be pecfe. A roaon abou he a repreene b h mar: R R A roaon abou he a repreene b h mar: A roaon abou he a repreene b h mar: R R R R D roaon can be vewe a replacng an wh wo ae.
8 Tng Yp Mah 8A 8 EXAMPLE Th wre polgon cube repreene b a mar ha conan vere pon n ever column. Roae Cube Orgnal Cube If we wan o roae h cube wh repec o he a b π : π π π π
9 Tng Yp Mah 8A Proecon Tranformaon Even hough we programme obec n -Dmenon, we have o acuall vew he obec a -Dmenon on our compuer creen. In anoher wor, we wan o ranform pon n R o pon n R. Parallel Proecon In parallel proecon, we mpl gnore he -coornae. Th operaon can be vewe a a ranformaon n homogeneou coornae,,, o,,,. Parallel Mar for Homogeneou Coornae n R 4 gven b h mar: P Gven an pon,, n R, he followng wll gve he parallel proece pon. Perpecve Proecon Veo game en o ue perpecve proecon over oher proecon o repreen a real worl, where parallelm no preerve. Perpecve Proecon he wa we ee hng,.e. bgger when he obec cloer. hp://mane.mech.vrgna.eu/~engr6/graphc/perpecve.hml
10 Tng Yp Mah 8A Imporan: Tranlae he ee o he Orgn Roaon unl recon of ee owar he negave -a D he ance of he ee o he vew plane he ance of he ee o he obec Noe: no our ee bu he ee of he compuer polgon peron Perpecve Mar for Homogenou Coornae n R 4 gven b h mar: Gven an pon,, n R, he followng wll gve he parallel proece pon. Noe: Th mar ranformaon oe no gve pel coornae on he monor. The ranforme coornae wh repec o he obec coornae. We have o ranlae he obec coornae o pel coornae on he monor. Ee Obec Perpecve Proecon of he cube
11 Tng Yp Mah 8A Graphc Screenho aen from Operaon Flahpon In npng moe, he ee move cloer o he obec. Concluon I choe o o h proec o how m curo n mah an compuer cence. I ha he chance o al abou veo game an mah ha are ofen overlooe a unrelae. A hown n h proec, Lnear Algebra eremel ueful for veo game graphc. Ug marce o manpulae pon a common mahemacal approach n veo game graphc.
12 Tng Yp Mah 8A Reference Dam, Anre. Inroucon o Compuer Graphc hp:// Holchuch, Ncola. Proecon an Perpecve hp:// La, Dav. Lnear Algebra an Applcaon. Secon Eon. Runwal, Rachana. Perpecve Proecon hp://mane.mech.vrgna.eu/~engr6/graphc/perpecve.hml
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