Geometry Transformation

Transcription

1 Geomer Transformaion Januar 26 Prof. Gar Wang Dep. of Mechanical and Manufacuring Engineering Universi of Manioba

2 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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4 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

5 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

6 2-D Transformaion Translaion V (, ) V(, ) d d o 6

7 Translaion 2-D Transformaion Ever eni of a geomeric model remains parallel o is iniial posiion V(, ) V (, ) d d + d d o 7

8 2-D Transformaion V (, ) Roaion V(, ) o φ 8

9 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

10 2-D Transformaion Scaling V(, ) V (, ) o

11 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

12 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

13 3 3-D Transformaion Translaion Translae poin V(,, ) b (d, d, d) o poin V (,, ) + d d d o d d d V V }

14 4 3-D Transformaion Scaling s s s o V V

15 5 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou Z

16 6 3-D Transformaion Roaion abou Roaion abou Z (eamples) Z (eamples)

17 7 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou X

18 8 V (, ) V(, ) φ o o + 3-D Transformaion Roaion abou Roaion abou Y

19 9 P 2 P 2 2 P P P P 3-D Transformaion Reflecion Reflecion

20 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

21 Homogeneous Represenaion The represenaion is inroduced o epress all geomeric ransformaions in he from of mari muliplicaion for he convenience of manipulaion. T T T 3 T 2 2

22 22 Homogeneous Represenaions ] [ s s s H Scaling Roaion [ ] [ ] 2 R and R P P

23 23 Homogeneous Represenaions [ ] [ ] ± ± ± 2 M and M P P [ ] [ ] ; R R Reflecion

24 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

25 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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27 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 T ( 3) ( 3) R ( 3) ( 3) T

28 P The definiion of he poin mari in he homogeneous represenaion. VT2*R*T*P V

29 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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