Systems & Biomedical Engineering Department. Transformation
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1 Sem & Biomedical Engineering Deparmen SBE 36B: Compuer Sem III Compuer Graphic Tranformaion Dr. Aman Eldeib Spring 28 Tranformaion Tranformaion i a fundamenal corner one of compuer graphic and i a cenral o OpenGL a well a mo oher graphic em.(2d and 3D Tranformaion Given an objec, ranformaion i o change he objec oiion (ranlaion Sie (caling Orienaion (roaion Shape (hear
2 2 Tranlaion Tranlae individual verice We can ranlae or move poin o a new poiion b adding offe o heir coordinae We can ranlae or move poin o a new poiion b adding offe o heir coordinae Tranlae individual verice D Con. Tranlaion
3 Homogeneou Coordinae Homogeneou coordinae: repreening coordinae in 2 dimenion wih a 3-vecor and coordinae in 3 dimenion wih a 4-vecor (,, w (Noe ha picall w in objec coordinae Homogeneou Coordinae Homogeneou coordinae eem uninuiive, bu he make graphic operaion much eaier Our ranformaion marice are now 44 for 3D T T T T Tranlaion mari T 3
4 4 We can ranlae or move poin o a new poiion b adding offe o heir coordinae Tranlae individual verice 3D T Con. Tranlaion Tranlae individual verice We can ranlae or move poin o a new poiion b adding offe o heir coordinae To ranform an objec mulipl each vere b he ame mari A quare ranlaion can be done a follow: ] [ ] [ T Con. Tranlaion
5 Scaling Scaling a coordinae mean mulipling each of i componen b a calar Uniform caling mean hi calar i he ame for all componen 2 Scaling Con. Scaling a coordinae mean mulipling each of i componen b a calar 2 5
6 6 Scaling a coordinae mean mulipling each of i componen b a calar 2D 3D S Scaling mari Con. Scaling Scaling a coordinae mean mulipling each of i componen b a calar Non-uniform caling: differen calar per componen X 2, Y.5 Con. Scaling
7 Fied oin Scaling Fied oin Scaling Con. 7
8 8 Fied oin Scaling Con. Sep : Tranlae o he origin Fied oin Scaling Con.
9 9 Sep 2: Scale he objec Fied oin Scaling Con. Sep 3: Tranlae he objec back Fied oin Scaling Con.
10 Fied oin Scaling Con.. Tranlae o origin 2. Scale 3. Tranlae back p ( T S T p 2D Roaion (, (, co( - in( in( + co( co in ( in( ( co(
11 3-D i more complicaed Need o pecif an ai of roaion There are four wa o pecif a 3D roaion Simple cae: roaion abou X, Y, Z ae 3-D roaion mari look like Roaion abou he X-ai co( in( in( co( Roaion abou he Y-ai Roaion abou he Z-ai co( in( in( co( co( in( in( co( Roaion Con. 3D Roaion abou he X-ai Roaion abou he Y-ai Roaion abou he Z-ai co in in co ( R co in in co ( R co in in co ( R 3-D i more complicaed Need o pecif an ai of roaion There are four wa o pecif a 3D roaion Simple cae: roaion abou X, Y, Z ae 3-D roaion mari look like Roaion Con. 3D
12 3D Roaion 3-D i more complicaed Need o pecif an ai of roaion There are four wa o pecif a 3D roaion 3D roaion ai angle repreenaion 3-D roaion mari look like 2 2 R( a, co + ( co + in 2 Roae a poin abou an arbirar ai a (,, going hrough he origin Noe: he ai a hould be of uni lengh a Con. Fied oin Roaing. Tranlae o origin 2. Roae 3. Tranlae back p ( T R T p 2
13 Mari Compoiion Mari muliplicaion doe no commue Tranformaion produc ma no be commuaive AB BA Shearing Y coordinae are unaffeced, bu coordinae are ranlaed linearl wih, i.e. + * h h 3
14 Shearing Con. X coordinae are unaffeced, bu coordinae are ranlaed linearl wih, i.e. + * g g Invere Tranformaion Tranlae Scale Shear Roae 4
15 OpenGL: Modeling Tranformaion OpenGL provide everal command for performing modeling ranform: gltranlae{fd}(,, Creae a mari T ha ranform an objec b ranlaing (moving i b he pecified,, and value glroae{fd}(angle,,, Creae a mari R ha ranform an objec b roaing i counerclockwie angle degree abou he vecor {,, } glscale{fd}(,, Creae a mari S ha cale an objec b he pecified facor in he,, and direcion OpenGL: Mari Manipulaion Each of hee po muliplie he curren mari E.g., if curren mari i C, hen CCS The curren mari i eiher he modelview mari or he projecion mari (alo a eure mari, won dicu for now Se hee wih glmarimode(, e.g.: glmarimode(gl_modelview; glmarimode(gl_rojection; WARNING: common miake ahead! Be ure ha ou are in GL_MODELVIEW mode before making modeling or viewing call! Ugl miake becaue i can appear o work, a lea for a while 5
16 OpenGL: Mari Manipulaion Con. More mari manipulaion call To replace he curren mari wih an ideni mari: glloadideni( omulipl he curren mari wih an arbirar mari: glmulmari{fd}(floa/double m[6] Cop he curren mari and puh i ono a ack: gluhmari( Dicard he curren mari and replace i wih whaever on op of he ack: glopmari( Noe ha here are mari ack for boh modelview and projecion mode OpenGL: Mari Manipulaion Con. glmarimode (GL_MODELVIEW glloadideni ( ; glmulmarif (m2; M M 2 M glmulmarif (m; OpenGL ue po muliplicaion when mulipling marice hu, ranformaion are applied in he invere order. The la one pecified i he fir one applied. 6
17 OpenGL: Mari Manipulaion Con. glmarimode (GL_MODELVIEW glloadideni ( ; glroaionf (hea,,,; drawcube ( ; gltranlaef (a,b,c; drawcube ( ; OpenGL: Mari Manipulaion Con. gltranlaef(a,b,c; T gluhmari ( ; glroaef(hea,a2,b2,c2; glscale(a3,b3,c3; T R 2 T R 2S3 glopmari ( ; gltranlaef(a4,b4,c4; T T 4 7
18 OpenGL: Tranformaion Hierarchie gluhmari(; // ranlae o houlder poiion // roae b houlder join // draw houlder (circle and recangle gluhmari(; // ranlae o elbow poiion // roae b elbow joing // draw elbow (circle and recangle glopmari(; glopmari(; OpenGL: Specifing Color Can pecif oher properie uch a color To produce a ingle aqua-colored riangle: glcolor3f(.,.5,.; glvere3fv(v; glvere3fv(v; glvere3fv(v2; To produce a Gouraud-haded riangle: glcolor3f(,, ; glvere3fv(v; glcolor3f(,, ; glvere3fv(v; glcolor3f(,, ; glvere3fv(v2; In OpenGL, color can alo have a fourh componen α (opaci Generall wan α. (opaque; 8
Computer Graphics. Transformation
(SBE 36) Dr. Aman Eldeib Spring 2 SBE 36 i a fundamental corner tone of computer graphic and i a central to OpenGL a well a mot other graphic tem.(2d and 3D ) Given an object, tranformation i to change
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