3D Coordinate Systems. 3D Geometric Transformation Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker. Right-handed coordinate system:

Transcription

1 3D Geomeric ransformaion Chap. 5 in FVD, Chap. in Hearn & Baker 3D Coordinae Ssems Righ-handed coordinae ssem: Lef-handed coordinae ssem:

2 2 Reminder: Vecor rodc U V UV VU sin ˆ V nu V U V U ˆ ˆ ˆ 3D oin A 3D poin is represened in homogeneos coordinaes b a 4-dim. ecor: Noe, ha α α α α p

3 3 3D ransformaions In homogeneos coordinaes, 3D Affine ransformaions are represened b 44 marices: A poin ransformaion is performed: i h g f e d c b a i h g f e d c b a Gien k poins (, 2,.. k ) ha hae been ransformed o (, 2,.., k ) b he 3D affine ransformaion. Ho man poins niqel define he ransformaion? Ho can e find he ransformaion? A i h g f e d c b a

4 4 3D ranslaion in ranslaed o b: Inerse ranslaion: Or Scaling c b a c b a S Or S

5 5 3D Reflecion A reflecion hrogh he plane: Reflecions hrogh he and he planes are defined similarl. Ho can e reflec hrogh some arbirar plane? 3D Shearing Shearing: he change in each coordinae is a linear combinaion of all hree. ransforms a cbe ino a general parallelepiped. f e d c b a f e d c b a

6 3D Roaion o generae a roaion in 3D e hae o specif: ais of roaion (2 d.o.f) amon of roaion ( d.o.f) Noe, he ais passes hrogh he origin. A coner-clockise roaion abo he -ais: cos sin sin cos p R ( ) p 6

7 7 A coner-clockise roaion abo he -ais: cos sin sin cos p R p ) ( A coner-clockise roaion abo he -ais: cos sin sin cos p R p ) (

8 Inerse Roaion p R ( ) p R( ) p R ( ) Composie Roaions R, R, and R, can perform an roaion abo an ais passing hrogh he origin. Composiion of ransformaions Rigid ransformaion: ranslaion Roaion (disance presering). Similari ransformaion: ranslaion Roaion niform Scale (angle presering). Affineransformaion: ranslaion Roaion Scale Shear (parallelism presering). All aboe ransformaions are grops here Rigid Similari Affine. 8

9 Changing Coordinae Ssems Affine Similari Rigid roblem: Gien he XYZ orhogonal coordinae ssem, find a ransformaion, M, ha maps a represenaion in XYZ o represenaion in orhogonal ssem UVW, ih he same origin. he mari M ransforms he UVW ecors o he XYZ ecors. (,, ) (,, ) (,, ) 9

10 Solion: M is roaion mari hose ros are U,V, and W: Noe, he inerse ransformaion is he ranspose: M M M Les check he ransformaion of U nder M: Similarl, V goes ino Y, and W goes ino Z. X MU 2 2 2

11 U M Les check he ransformaion of he X ais nder M - : Similarl, Y goes ino V, and Z goes ino W. Roaion Abo an Arbirar Ais Ais of roaion can be locaed a an poin: 6 d.o.f. he idea: make he ais coinciden ih one of he coordinae aes ( ais), roae b, and hen ransform back. Assme ha he ais passes hrogh he poins p and p 2. p p 2

12 p p 2 Iniial osiion sep 3 p 2 p Roae he Objec Arond he Ais Consrcing an orhogonal ssem along he roaion ais: A ecor W parallel o he roaion ais: sep p p 2 p p 2 ranslae p o he Origin sep 2 Roae p 2 ono he Ais sep 4 sep 5 p p 2 Roae he Ais o he original Orienaion p p 2 ranslae o he Original osiion p2 p ; s s s A ecor V perpendiclar o W: A ecor U forming a righ-handed orhogonal ssem ih W and V: 2

13 3 Sep: Sep2: Sep3: M cos sin sin cos R Sep4: Sep5: M RM M Roaing abo an ais:

14 ransforming lanes lane represenaion: B hree non-collinear poins B implici eqaion: A B C D [ A B C D] One a o ransform a plane is b ransforming an hree noncollinear poins on he plane. Anoher a is o ransform he plane eqaion: Gien a ransformaion ha ransforms [,,,] o [,,,] find [A,B,C,D], sch ha: A D B D C D 2 [ ] [ A B C D ] 4

15 5 Noe ha hs, he ransformaion ha e shold appl o he plane eqaion is: ( ) D C B A ( ) ( ) D C B A D C B A