Geomer Transformaion Januar 26 Prof. Gar Wang Dep. of Mechanical and Manufacuring Engineering Universi of Manioba
Wh geomer ransformaion? Beer undersanding of he design Communicaion wih cusomers Generaing various oupus Common ransformaions: Translaion Roaion Scaling Reflecion 2
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Geomeric Transformaion I can change he descripion of a geomeric model of an objec in a coordinae ssem. I can map he coordinae values of an objec from one coordinae ssem o anoher. Translaion, Uniform Scaling, Roaion, and Reflecion (Mirror) 4
Righ-Handed Convenion A roaion angle abou a given ais is posiive in a counerclockwise sense when viewed form a poin on he posiive porion of he ais oward he origin. Y Transformaion of a poin Z X Given a poin P ha belongs o a geomeric model, find he corresponding poin P * in he new posiion such ha P * [T]P 5
2-D Transformaion Translaion V (, ) V(, ) d d o 6
Translaion 2-D Transformaion Ever eni of a geomeric model remains parallel o is iniial posiion V(, ) V (, ) d d + d d o 7
2-D Transformaion V (, ) Roaion V(, ) o φ 8
9 P (, ) P(, ) φ o 2-D Transformaion Roaion Roaion + + + + r r r r P r r P φ φ φ φ φ φ φ φ r r ) ( ) ( r
2-D Transformaion Scaling V(, ) V (, ) o
Scaling 2-D Transformaion Scaling is used o increase or decrease he sie of an eni Uniform scaling: s s s s. The model changes in sie onl and no in shape. o V(, ) V (, ) s s
2-D Transformaion An Eample In 2-D space, deermine he new posiion of poin A(, 5). A is ranslaed a disance of 3 unis along posiive X direcion and hen roaed 3 degree clockwise abou he origin O (or ais in 3-D space). o 2
3 3-D Transformaion Translaion Translae poin V(,, ) b (d, d, d) o poin V (,, ) + d d d o d d d V V }
4 3-D Transformaion Scaling s s s o V V
5 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou Z
6 3-D Transformaion Roaion abou Roaion abou Z (eamples) Z (eamples)
7 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou X
8 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou Y
9 P 2 P 2 2 P P P P 3-D Transformaion Reflecion Reflecion
2 Homogeneous Represenaion The represenaion is inroduced o epress all geomeric ransformaions in he from of mari muliplicaion for he convenience of manipulaion. Dumm (n+)h coordinae o faciliae muliplicaion d d d
Homogeneous Represenaion The represenaion is inroduced o epress all geomeric ransformaions in he from of mari muliplicaion for he convenience of manipulaion. T 2 3 4 2 22 32 42 3 23 33 43 4 24 34 44 T T 3 T 2 2
22 Homogeneous Represenaions ] [ s s s H Scaling Roaion [ ] [ ] 2 R and R P P
23 Homogeneous Represenaions [ ] [ ] ± ± ± 2 M and M P P [ ] [ ] ; R R Reflecion
Composie/Concaenaed Transformaion V [ H ][ H ] [ H ] V n n The order of ransformaions does maer in general Translae Roae 2 Roae 2 Translae 24
An Eample Consider a 3D objec. The coordinaes of he verices are given as follows: A[3, 5, 3] B[7, 5, 3] C[7, 5, 5] D[3, 5, 5] E[3, 6, 5] F[3, 6, 3] Roae he 3D objec b 3 degree in clockwise (CW) direcion a poin D abou he -ais. 25
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Procedure: Firs we ranslae (T) he objec a he refernece poin D o he origin. Then we roae (R) abou he -ais Finall, we ranslae (T2) he poin D from he origin back o is original posiion. 3 5 5 T ( 3) ( 3) R ( 3) ( 3) T 2 3 5 5 27
P 3 7 7 3 3 3 5 5 5 5 6 6 3 3 5 5 5 3 The definiion of he poin mari in he homogeneous represenaion. VT2*R*T*P V 4. 7.46 6.46 3. 3. 4. 5. 5. 5. 5. 6. 6. 3.27 5.27 7. 5. 5. 3.27...... 28
Think How is geomeric ransformaion applied in making movies (e.g. Finding Nemo)? How is Gollum creaed in he Lord of he Rings? (Hin: moion capure) 29
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