4.1 3D GEOMETRIC TRANSFORMATIONS

Transcription

1 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- Dep. of Compuer Science And Applicaions, SJCET, Palai D GEOMETRIC TRANSFORMATIONS Mehods for geomeric ransformaions and objec modeling in hree dimensions are eended from wo-dimensional mehods b including consideraions for he coordinae. We now ranslae an objec b specifing a hree-dimensional ranslaion vecor, which deermines how much he objec is o be moved in each of he hree coordinae direcions. Similarl, we scale an objec wih hree coordinae scaling facors. The eension for hree-dimensional roaion is less sraighforward. When we discussed wo-dimensional roaions in he plane, we needed o consider onl roaions abou aes ha were perpendicular o he plane. In hree-dimensional space, we can now selec an spaial orienaion for he roaion ais. Mos graphics packages handle hreedimensional roaion as a composie of hree roaions, one for each of he hree Caresian aes. Alernaivel, a user can easil se up a general roaion mari, given he orienaion of he ais and he required roaion angle. As in he wo-dimensional case, we epress geomeric ransformaions in mari form. An sequence of ransformaions is hen represened as, a gle mari, formed b concaenaing he marices for he individual ransformaions in he sequence. Transformaion Mari 33 : Scaling, Reflecion, Shearing, Roaion 3 : Translaion : Uniform global Scaling 3 : Homogeneous represenaion 4.. TRANSLATION In a hree-dimensional homogeneous coordinae represenaion, a poin is ranslaed (Fig. 4.) from posiion P = (,, ) o posiion P = (,, ) wih he mari operaion S L I F C K H E B J G D A,,

2 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- (Fig: 4.) An objec is ranslaed in hree dimensions b ransforming each of he defining poins of he objec. For an objec represened as a se of polgon surfaces, we ranslae each vere of each surface and redraw he polgon faces in he new posiion. We obain he inverse of he ranslaion mari in he given equaion b negaing he ranslaion disances,, and. This produces a ranslaion in he opposie direcion, and he produc of a ranslaion mari and is inverse produces he ideni mari ROTATION To generae a roaion ransformaion for an objec, we mus designae an ais of roaion (abou which he objec is o be roaed) and he amoun of angular roaion. Unlike wo-dimensional applicaions, where all ransformaions are carried ou in he plane, a hree-dimensional roaion can be specified around an line in space. The easies roaion aes o handle are hose ha are parallel o he coordinae aes. Also, we can use combinaions of coordinae ais roaions (along wih appropriae ranslaions) o specif an general roaion. B convenion, posiive roaion angles produce counerclockwise roaions abou a coordinae ais, if we are looking along he posiive half of he ais oward he coordinae origin his agrees wih our earlier discussion of roaion in wo dimensions, where posiive roaions in he plane are counerclockwise abou aes parallel o he ais. Coordinae-Aes Roaions X-ais roaion Y-ais roaion Z-ais roaion Dep. of Compuer Science And Applicaions, SJCET, Palai 95

3 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- Dep. of Compuer Science And Applicaions, SJCET, Palai 96 X-Ais Roaion Z-Ais Roaion Y-Ais Roaion An inverse roaion mari is formed b replacing he roaion angle. Negaive values for roaion angles generae roaions in a clockwise direcion, so he ideni mari is produced when an roaion mari is muliplied b is inverse. Since onl he e funcion is affeced b he change in sign of he roaion angle, he inverse mari can also be obained b inerchanging rows and columns. Tha is, we can calculae he inverse of an roaion mari R b evaluaing is ranspose (R - = R T ). This mehod for obaining an inverse mari holds also for an composie roaion mari General Three-Dimensional Roaions A roaion mari for an ais ha does no coincide wih a coordinae ais can be se up as a composie ransformaion involving combinaions of ranslaions and he coordinae-aes roaions. We obain he required composie mari b firs seing up he ransformaion sequence ha moves he seleced roaion ais ono one of he coordinae aes. Then we se up he roaion mari abou ha coordinae ais for he specified roaion angle. The las sep is o obain he inverse ransformaion sequence ha reurns he roaion ais o is original posiion.

4 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- In he special case where an objec is o be roaed abou an ais ha is parallel o one of he coordinae aes, we can aain he desired roaion wih he following ransformaion sequence. ) Translae he objec so ha he roaion ais coincides wih he parallel coordinae ais. 2) Perform he specified roaion abou ha ais. 3) Translae he objec so ha he roaion ais is moved back o is original posiion. The seps in his sequence are illusraed in Fig An coordinae posiion P on he objec in his figure is ransformed wih he sequence shown as P = -.R (θ).t.p Where he composie mari for he ransformaion is R (θ) =T -. R (θ).t This is of he same as he wo-dimensional ransformaion sequence for roaion abou an arbirar pivo poin. (Fig : 4.2) Dep. of Compuer Science And Applicaions, SJCET, Palai 97

5 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- When an objec is o be roaed abou an ais ha is no parallel o one of he coordinae aes, we need o perform some addiional ransformaions. In his case, we also need roaions o align he ais wih a seleced coordinae ais and o bring he ais back o is original orienaion Given he specificaions for he roaion ais and he roaion angle, we can accomplish he required roaion in five seps ) Translae he objec so ha he roaion ais pass= hrough he coordinae origin. 2) Roae he objec so ha he ais of roaion coincides wih one of he coordinae aes. 3) Perform he specified roaion abou ha coordinae ais. 4) Appl inverse roaions o bring he roaion ais back o is original orienaion. 5) Appl he inverse ranslaion o bring he roaion ais back o is original posiion. Dep. of Compuer Science And Applicaions, SJCET, Palai 98

6 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- Dep. of Compuer Science And Applicaions, SJCET, Palai 99

7 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- Dep. of Compuer Science And Applicaions, SJCET, Palai

8 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN SCALING The mari epression or he scaling ransformaion of a posiion P = (,, ) relaive o he coordinae origin can be wrien as Where scaling parameers s, s, and s are assigned an posiive values. Eplici epressions for he coordinae ransformaions for scaling relaive o he origin are Scaling wih respec o a seleced fied posiion (,, ) can be represened wih he following ransformaion sequence: ) Translae he fied poin o he origin. 2) Scale he objec relaive o he coordinae origin. 3) Translae he fied poin back o is original posiion. Dep. of Compuer Science And Applicaions, SJCET, Palai

9 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN OTHER TRANSFORMATIONS In addiion o ranslaion, roaion, and scaling, here are various addiional ransformaions ha are ofen useful in hree-dimensional graphics applicaions. Two of hese are reflecion and shear REFLECTIONS A hree-dimensional reflecion can be performed relaive o a seleced reflecion ais or wih respec o a seleced reflecion plane. In general, hree-dimensional reflecion marices are se up similarl o hose for wo dimensions. Reflecions relaive o a given ais are equivalen o 8 o roaions abou ha ais. Reflecions wih respec o a plane are equivalen o 8 o roaions in four-dimensional space. When he reflecion plane is a coordinae plane (eiher,, or ), we can hink of he ransformaion as a conversion beween Lef-handed and righ-handed ssems. An eample of a reflecion ha convers coordinae specificaions from a righhanded ssem o a lef-handed ssem (or vice versa) is shown in Fig This ransformaion changes he sign of he coordinaes, Leaving he and -coordinae values unchanged. The mari represenaion for his reflecion of poins relaive o he plane is given below Dep. of Compuer Science And Applicaions, SJCET, Palai 2

10 MODULE IV MCA - 3 COMPUTER GRAPHICS ADMN 29- (Fig: 4.3) Transformaion marices for invering and values are defined similarl, as reflecions relaive o he plane and plane, respecivel Reflecions abou oher planes can be obained as a combinaion of roaions and coordinae-plane reflecions SHEARS Shearing ransformaions can he used o modif objec shapes. The are also useful in hree-dimensional viewing for obaining general projecion ransformaions. In wo dimensions, we discussed ransformaion relaive o he or aes o produce disorions in he shapes of objecs. In hree dimensions, we can also generae shears relaive o he ais. As an eample of hree-dimensional shearing he following ransformaion produces a -ais shear: (Fig: 4.4) Parameers a and b can be assigned an real values. The effec of his ransformaion mari is o aler - and -coordinae values b an amoun ha is proporional o he value, while leaving he coordinae unchanged. Boundaries of planes ha are perpendicular o he ais are hus shifed b an amoun proporional o. An eample of he effec of his shearing mari on a uni cube is shown in Fig. 4-4, for shearing values a = b =. Shearing marices for he ais and ais are defined similarl. Dep. of Compuer Science And Applicaions, SJCET, Palai 3