Geometric transformations

Transcription

1 Geometric trnsformtions Computer Grphics Some slides re bsed on Shy Shlom slides from TAU

2 mn n n m m T A,,,,,, Rows become columns nd columns become rows nm n n m m A,,,,,,

3 Mtrices A nd B hve these dimensions: [r x c] nd [s x d] Mtrices A nd B cn be multiplied if: c = s The resulting mtrix will hve the dimensions: r x d

4 A B b11 b12 b13 b21 b22 b23 [2 x 2] [2 x 3] C b b b b b b b b [2 x 3] b b b b 23 23

5 A nd B [3 x 2] [2 x 3] A nd B cn be multiplied C 2*1 3*1 5 1*11*1 2 1*1 0*1 1 2*1 3*0 2 1*11*0 1 1*1 0*0 1 2*1 3*2 8 1*11*2 3 1*1 0* [3 x 3]

6 A nd B D 1*2 1*11*1 4 1*2 0*1 2*1 4 1*31*11*0 4 1*3 0*1 2*0 [2 x 2] D C

7 Estblish your position in the scene Position objects within the scene Scle objects Estblish perspective trnsformtion Moving round in OpenGL using cmer 7

8 Tnrsformtion Viewing Modeling ModelView Projection Viewport Use Specifies the loction of the viewer or cmer Moves objects round the scene Describes the dulity of viewing nd modeling trnsformtions Clips nd sizes the viewing volume Scles the finl output to the window 8

9 Viewpoint of the observer They re virtul fixed coordinte system Used s common frme of reference 9

10 Like plcing nd pointing the cmer t the scene To be specified before ny other trnsformtion By defult, the point of observtion is t the origin, looking down the negtive Z-xis. Wht bout objects drwn with positive Z vlues? 10

11 Mnipulte the model nd the prticulr objects within it Move objects into plce, rotte them, nd scle them These re the most common modeling trnsformtions 11

12 Finl ppernce of n object depends gretly on the order of trnsformtions 12

13 In fct, viewing nd modeling trnsformtions re the sme Induce the sme effects on the finl ppernce of the scene The distinctions is mde s convenience for the progrmmer 13

14 glortho(left, right, bottom, top, -ner, -fr) 14

15 glfrustum(left, right, bottom, top, ner, fr) gluperspective(gldouble fov,gldouble spectrtio,gldouble zner,gldouble zfr) 15

16 Applied fter the modelview trnsformtion Defines the viewing volume nd the clipping plnes Specifies how finished scene is projected to the screen 16

17 In n Orthogrphic projection: objects re mpped directly on the 2D screen using prllel lines No mtter how fr wy something is In Perspective projection: Scenes re more relistic Distnt objects pper smller thn nerby objects of the sme size Prllel lines in 3D spce do not lwys pper prllel See Code Exmple 3 17

18 18

19 glulookat(gldouble eyex, GLdouble eyey, Gldouble eyez, GLdouble centerx, GLdouble centery, GLdouble centerz, GLdouble upx, GLdouble upy, GLdouble upz) 19

20 Current Mtrix is prt of the OpenGL stte Options re: GL_MODELVIEW chnges, rotte, trnslte the model GL_PROJECTION chnges, trnslte, rotte the cmer To specify which one is the Current Mtrix : glmtrixmode( ) Subsequent mtrix opertions ffect the specified mtrix One of this mtrices could be chnge in given time 20

21 The defult mtrix is the identity mtrix: We cn chnge the current mtrix to the identity mtrix ny time use: gllodidentity(); 21

22 void gltrnsltef(glflot x, GLflot y, GLflot z ) This function tkes s prmeters the mount to trnslte long the x, y, nd z directions 22

23 gltrnslte(0.0f, 10.0f, 0.0f) glutwirecube(10.0f) Construct trnsltion mtrix for positive 10 Y Multiply it by the modelview mtrix 23

24 void glrottef(glflot ngle,glflot x, GLflot y, GLflot z ) This function performs rottion round the vector specified by the x, y, nd z rguments. The ngle of rottion is in the counterclockwise direction. A rottion round n rbitrry xis could be done. 24

25 void glsclef(glflot x, GLflot y, GLflot z ) Multiplies the x, y nd z vlues by the scling fctors specified. Scling does not hve to be uniform. 25

26 Mtrix opertions re cumultive Erroneous rtifcts my occur to the scene You reset the origin by loding the modelview mtrix with the identity mtrix gltrnsltef(0.0f, 10.0f, 0.0f); glutsolidsphere(1.0f,32,32); gltrnsltef(10.0f, 0.0f, 0.0f); glutsolidsphere(1.0f) 26

27 glmtrixmode(gl_modelview) gllodidentity(); gltrnsltef(0.0f, 10.0f, 0.0f); glutsolidsphere(1.0f,32,32); gllodidentity(); gltrnsltef(10.0f, 0.0f, 0.0f); glutsolidsphere(1.0f) 27

28 Resetting the modelview mtrix to identity before plcing the every object is not lwys desirble To sve the current trnsformtion stte OpenGL mintins mtrix stck for both the modelview nd projection mtrices glpushmtrix() push the current stte mtrix to the stck glpopmtrix() pop the top mtrix out of the stck, nd replce the current stte mtrix with it 28

29 See Code Exmple 4 29

30 We cn define flot rry nd lod the n OpenGL mtrix to it: flot modelmtrix[16],projectionmtrix[16]; glgetflotv(gl_projection_matrix, projectionmtrix); glgetflotv(gl_modelview_matrix, modelmtrix); We cn chnge it nd lod it bck insted of the current mtrix: gllodmtrixf(modelmtrix); 30

31 After loding mtrix to n rry the indices of the rry which represent ech vlue in the mtrix will be s follow:

32 In some cses we wnt to crete our 4x4 mtrix nd multiply it by nother mtrix or multiply OpenGL mtrix by our mtrix. glmultmtrixf(selfmtrix); Remember ll the opertion we mentioned on this P.S. multiply the current mtrix by nother mtrix! 32

33 Z y Euler Angles, definition: A set of three ngles used to describe the orienttion of reference frme in 3D spce Drws on the following observtion: You cn lign (superimpose) the globl reference frme to ny rbitrry reference frme through sequence of THREE rottion opertions z X x Y In picture: strt from blue RF, end in the red RF fter three rottions The sequence of three rottions tht we ll consider is About xes Z then X then Z gin (clled the sequence) We ll denote the three Euler Angles by φ, θ, nd ψ, respectively 33

34 Rotte round z xis Rotte round new x xis Rotte round new z xis. 34

35 Using the expression of the mtrices A 1, A 2, A 3, one gets for the expression of the orienttion mtrix A Crry out multiplictions to get 35

36 The cmer is the first object to be locted in (0,0,0) You cn drw ech object in scene coordinte but then you hve to clculte rottion nd position by yourself. The better option is to drw ech object when its center is on the origin nd fterwrd use rottion nd trnsltion to move it to its position in the scene. Remember: Rottion of OpenGL is lwys round the origin! 36

37 Use gllodidentity () t the initil prt nd not in other function. unless you wnt to clculte ech time ll trnsformtions Remember tht rottion nd trnsltion is ccumulted Rotting chnge your xis system Build for ech object (include scene) its own modelview mtrix Remember tht the order of trnsformtion ffects the result. 37