5. Geometric Transformations and Projections

Transcription

1 5. Geometic Tansfomations and ojections 5. Tanslations and Rotations a) Tanslation d d d d d d d d b) Scaling s s s s c) Reflection (about - - lane)

2 d) Rotation about Ais ( ) ( ) CCW 5.. Homogeneous Repesentation The epesentation is intoduced to epess all geometic tansfomations in the fom of mati multiplication fo the convenience of manipulation. a) Tanslation [ D] o T T T T T T b) Scaling S S [ S] and [ S] S c) Reflection (Mioing) [ M] and [ M] ± ± ± - - Compute Aided Design

3 - 3 - Compute Aided Design 3 d) Rotation about the Ais R [ ] and R [ ] Similal R [ ] ; R [ ] Rotation about Z Ais - CCW b Z X Y ( ) ( ) [ ] R

4 - 4 - Compute Aided Design 4 Rotation about X Ais - CCW b X Y Z ( ) ( ) [ ] R Rotation about Y Ais - CCW b Z X Y

5 - 5 - Compute Aided Design 5 ( ) ( ) [ ] R Rotation about an Abita Ais aametic Rep. of the Ais: Au Bu Cu L A B C u < u < Step : Tanslate to Oigin D [ ] X Y Z X X Y Z L L A B C L

6 6 Step : Rotate Vecto v L about X Ais to get v L into the - plane L A B C V B C B V C V [ R ] C V B V B C V V Step 3: Rotate v L about the Y ais to get it in the diection Rotate a negative angle (CW)! Compute Aided Design

7 A L V L co V L A L [ R ] A L V L Step 4: Rotate angle about ais v L. [ R ] Step 5: Revese the otation about the Y ais [ R ] Invese of Rotation: Replace b V A L L A L V L b b Step 6: Revese otation about the X ais - - Compute Aided Design

8 3 [ R ] C B V V B C V V Step 7: Revese tanslation Oveall Tansfomation [ D] Homewok: [ T] [ D] [ R ] [ R ] [ R ][ R ] R [ T ]. Give eal X, Y, Z, A, B, C and theta values, find [T]. [ ] D [ ]. A line connects the point A at (,,) to the point B at (,,). A second line etends fom Cat (,,) to D at (,,). Rotate line AB about line CD ug vecto-mati methods. The otation should be 9 counte-clockwise as seen fom the ais. 3. A plane suface intesects the coodinate aes at thee points A(5,,), B(,5,) and C(,,l). A given point is on the plane. Find the mati of geometic tansfomation that move the point five units down on the plane to '. (Line ' is pependicula to edge AB and is on plane ABC. '5.) Compute Aided Design

9 4 5.3 Viewing Coodinate Sstem a) View Window Right hand coodinate sstem D coodinate sstem on the view plane. Some "old" gaphics sstems use a left-hand coodinate sstem. A diffeent geometic tansfomation mati must be used Compute Aided Design

10 5 b) Othogaphic View vesus Wold Coodinate Sstem Compute Aided Design

11 6 c) espective and aallel ojections We want to poduce a D image (pojection) of a 3D object. espective ojection aallel ojection Eamples of espective ojection Compute Aided Design

12 7 5.4 Definition of a Geneal Viewing Coodinate Sstem a) What ae involved? b) How to Set Up the Viewing Coodinate Sstem (VCS)! i)define the view efeence point ii) ( ) T,, Define the line of the sight vecto n v (nomalied) v n N, N, N ( ) T and N N N iii) Define the "up" diection v V V, V, V ( ) T v n, v V v n This also defines an othogonal vecto, u v u V v v n u, V v, v n ( ) foms the viewing coodinates Compute Aided Design

13 iv) Define the View Window in v U v V v W coodinates 8 c) aallel ojection Fist tansfom coodinates of objects into the UVn coodinates (VCS), then dop the n component. (n depth) Ovelapping - - and U - V - n i) Tanslate O v to O. [ D] v v v ii) Align the v n ais with the Z ais. A N, B N, C N L N N N V N N The pocedue is identical to that given in 5.. otate about : [R ] otate about : [R ] Compute Aided Design

14 9 iii) Rotate 3 about the Z ais to align U with and/o V with. At this point, V is given b V V ( V, V, ) T whee [ R ][ R ] Do v,o [ ] V V V We need to otate b an angle 3 about the Z ais L V V, 3 V L, 3 V L V L V L V L V L [ R ] Compute Aided Design

15 iv) Dop the n coodinate [ D n ], u V [ ] D n u V n In summa, to poject a view of an object on the UV plane, one needs to tansfom each point on the object b: [ T] D n [ ][ R ] R u V [ ] R [ T] [ T] [ ][ Do v,o] Note: The invese tansfoms ae not needed! We don't want to go back to - - coodinates. - - Compute Aided Design