Lecture 8. 3D Geometric Transformation
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1 Lecture 8 3D Gemetric Trasfrmati
2 Applig trasfrmati matrices D D z z z z z z 3
3 Traslati D Traslati 3D Traslati TR TR z Cmpet-wise additi f vectrs = + d = + d z = z + dz T mve plgs: traslate vertices (vectrs ad redraw lies betwee them Preserves legths ad agles Des t require referece pit
4 Rtati Rtati D R cs si si cs (X cs - Y si, si + Y cs A A (X,Y Ceter f rtati (, Rtati f vectrs thrugh a agle = cs Ө si Ө = si Ө + cs Ө Preserves legths ad agles Pit f rtati pit (,
5 Rtati at ther pit tha rigi pit Let sa that the ceter f rtati is (,. Traslati has t be carried befre ad after rtati Prcedure. Cvert the (, t (, b applig traslati = - ad = -.. Appl the rtati matri 3. Cvert the (, t its rigial pit (, b applig traslati = ad =. R
6 3D Rtati cs si si cs X R cs si si cs Y R cs si si cs Z R Ais f rtati: Ais f rtati: z Ais f rtati: If ais f rtati ther tha specified ais: appl traslati befre ad after 3D rtati is applied
7 Fudati f trasfrmatis Similar methd ca be applied ther trasfrmati such as shear, reflecti etc. Imprtat, if the trasfrmati requires pit (D, ais (D & 3D ad plae (3D at the rigi CS. Traslati must be applied befre ad after applig the respective trasfrmati.
8 Applig trasfrmati cmpsite matrices D this trasfrmati scree. Draw rectagle a size. Traslate d = 6 3. Rtate it b 45. The same rectagle. Rtate it Traslate d = 6 Q: D bth rectagles verlappig each ther? Wh?
9 Cmpsite Trasfrmatis Y Traslati Rtati X Y Rtati Traslati X
10 Cmpsite Trasfrmati C... X X C 3 Cmpsite Trasfrmati: Cmbiati f multiple trasfrmatis must be i sequece.
11 Eample A cube ( as shw i Figure a is trasfrmed it the slid as shw i Figure b. Establish the trasfrmati matrices fr this trasfrmati. The scalig matri trasfrmati based referece pit f (,, is as i the fllwig. Z S S S S
12 Eample : Sluti 4 4 [Scalig][Traslati] Reame 3Dslidcube.cpp ad tr this prblem Nte:. Matri i pegl is applied i reverse sequece, last will be applied first ad first will be applied last.. B assig variable t the scale matri ad this variable will be ctrlled b the idle fucti, shw a aimati f the cube becmes bigger.
13 Eample The available API fucti allws l t draw cube with ( l l l size. Based this fucti ad the trasfrmati matrices, mdel the rbtic arm with arm f ( ad ( 5 at the psiti as shw the figure. The ceter bttm f the lwer arm is at (,,. Discuss the methdlg t mdel bth lwer ad upper arm t be psitied as i the figure
14 Step : Scalig t lwer ad upper * L arm L arm The scalig f lwer arm The scalig f the upper arm 5 * U arm U arm Eample
15 Step : Traslate the upper arm at the ed f the lwer arm * * U arm U arm Step 3: Rtate lwer ad upper arm alg +ve Y ais b 3 cs(3 si( 3 6 si( cs(3 * 3* L arm L arm The trasfrmati fr the lwer arm. The trasfrmati fr the upper arm 6 si( cs(6 cs(3 si( 3 si( 3 cs(3 6 si( cs(6 * 3* U arm U arm Eample
16 Step 4: Rtate upper arm alg +ve Y ais b 6 si( cs(6 cs( si( si( cs( 6 si( cs(6 3* 4* U arm U arm The ceter rtati is at the ed f the lwer arm which is ( cs(6, si(6. Therefre, traslati must be applied befre ad after the rtati matri. Eample
17 Step 5: Rtate the lwer ad upper arm b 5 alg +ve Z ais cs(5 si(5 si(5 cs(5 3* 5* L arm L arm The trasfrmati fr the lwer arm. The trasfrmati fr the upper arm. 6 si( cs(6 cs(5 si(5 si(5 cs(5 6 si( cs(6 4* 5* U arm U arm Eample
18 Eample 3: Plaet ad Su Yu have t d aimati f plaet ges arud su usig tw slid sphere. Reame 3Dcubeslid.cpp ad mdif the prgram t mdel the su ad the plaet usig sphere with 5 ad uit diameter respectivel. The plaet rbit is uit radius.
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