1/29/13. Computer Graphics. Transformations. Simple Transformations

Transcription

1 /29/3 Computer Graphics Transformations Simple Transformations

2 /29/3 Contet 3D Coordinate Sstems Right hand (or counterclockwise) coordinate sstem Left hand coordinate sstem Not used in this class and Not in OpenGL 2

3 /29/3 What is a Transformation? Maps point (,, ) in one coordinate sstem to point (',, ) in another coordinate sstem ' m + m 2 + m 3 +m 4 ' m 2 + m 22 + m 23 +m 24 m 3 + m 32 + m 33 +m 34 Affine transformation How are Transforms Represented? ' m + m 2 + m 3 +m 4 ' m 2 + m 22 + m 23 +m 24 m 3 + m 32 + m 33 +m 34 ' m m 2 m 3 m 2 m 22 m 23 m 3 m 32 m 33 + m 4 m 24 m 34 p' M p + t 3

4 /29/3 Homogeneous Coordinates Add an etra dimension ' ' ' m m 2 m 3 m 4 m 2 m 22 m 23 m 24 m 3 m 32 m 33 m 34 p' M p In General ' ' ' w m m 2 m 3 m 4 m 2 m 22 m 23 m 24 m 3 m 32 m 33 m 34 m 4 m 42 m 43 m 44 w p' M p Most of the time w and we can ignore it. 4

5 /29/3 Homogeneous to Cartesian (2D) Simpl divide b w: w? Point at infinit w is a scale factor Homogeneous Visualiation (2D) Divide b w to normalie (homogenie) (,, w) (,, ) (,, 2) (7,, ) (4, 2, 2) w (4, 5, ) (8,, 2) w 2 5

6 /29/3 2D Transformations Translate (t, t) ʹ + t ʹ + t Translate(c,) p p' ' ' t t T(t,t ) c OpenGL: gltranslated(t, t, ) Note: 3 rd dimension makes it possible to encode transla;ons in the matri! 6

7 /29/3 Translation of Objects How to translate an object with multiple vertices? Translate individual vertices " $ #!! % " ' $ & $ # s s Scale (s, s) % ' & ( ʹ, ʹ ) ' (, ) ' ' s s S(s,s ) fied point OpenGL: glscaled(s, s, ) 7

8 /29/3 8 Reflection vs. Scale Along X-ais Along Y-ais ʹ ʹ ʹ ʹ Rotation ' ' cos sin -sin cos OpenGL: glrotated(,,, ) ( ), ( ) ʹ ʹ, fied point ' cos sin ' sin + cos R()

9 /29/3 Fied Rotation Point Default rotation center is origin (,) > : Rotate counterclockwise < : Rotate clockwise Rotation How to rotate an object with multiple vertices? 9

10 /29/3 Deriving the Rotation Matri (,) - > Rotate about the origin b How to compute (, )? (, ) r cos () r sin () r cos (φ + ) r sin (φ + ) Deriving the Rotation Matri Use trigonometr identities cos( +φ) cos cos φ - sin sin φ sin( +φ) sin cos φ + cos sin φ Yields cos sin cos + sin

11 /29/3 Arbitrar Rotation Center Where is the rotation center here? Arbitrar Rotation Center To rotate about arbitrar point P ( r, r ) b : Translate object b (- r, - r ) so that P is at origin Rotate the object b Translate object back b ( r, r ) In matri form: T( r, r) R( ) T( r, r)

12 /29/3 2 Arbitrar Rota;on Center + sin ) cos ( cos sin sin ) cos ( sin cos cos sin sin cos ), ( ) ( ), ( r r r r r r r r r r r r T R T How are Transforms Combined? TS Scale then Translate Use matri multiplication: p' T ( S p ) TS p Caution: matri multiplication is NOT commutative! (,) (,) (2,2) (,) (5,3) (3,) Scale(2,2) Translate(3,)

13 /29/3 Non-Commutative Compositions Scale then Translate: p' T ( S p ) TS p (,) Scale(2,2) (2,2) Translate(3,) (3,) (5,3) (,) (,) Translate then Scale: p' S ( T p ) ST p (8,4) (,) (,) Translate(3,) (3,) (4,2) Scale(2,2) (6,2) Non-Commutative Compositions Scale then Translate: p' T ( S p ) TS p TS Translate then Scale: p' S ( T p ) ST p ST

14 /29/3 Hands-On Session 2D Transformation Applet at applets/transformations2d.html or biological_anthropolog/ _virtual_reconstruction/chapter5_trafo.html Request handout from the instructor 3D Transformations 4

15 /29/3 Translate (t, t, t) Note: 4 th dimension makes it possible to encode transla;ons in the matri! Translate(c,,) p p' c ' ' ' t t t T(t,t,t ) OpenGL: gltranslated(t, t, t ) Scale (s, s, s) Isotropic (uniform) scaling: s s s p p' q q' ' ' ' s s s S(s,s,s ) OpenGL: glscaled(s, s, s ) 5

16 /29/3 6 3D Reflec;on About the - plane: ʹ ʹ ʹ Rotation About ais ' ' ' cos sin -sin cos p p' OpenGL: glrotated(,,, )

17 /29/3 Rotation About ais p' p ' ' ' cos sin -sin cos OpenGL: glrotated(,,, ) Rotation About ais p p' ' ' ' cos -sin sin cos OpenGL: glrotated(,,, ) 7

18 /29/3 About Arbitrar Ais r (r,r,r ) Rotation Rotate(, r) r ' ' ' rr(-c)+c rr(-c)+rs rr(-c)-rs rr(-c)-rs rr(-c)+c rr(-c)-rs rr(-c)+rs rr(-c)-rs rr(-c)+c where c cos & s sin Rota;on About Arbitrar Ais Rotate b about the ais (r, r, r ). Strateg: Align the rota;on ais with the ais (two rota;ons) Rotate b around Undo the two alignment rota;ons (r, r, r ) R R (- α ) R (- α ) R () R (α ) R (α ) Problem: Finding α, α 8

19 /29/3 9 Computing the -Rotation α Project unit vector (r, r, r ) onto plane d r 2 + r 2, cos α r / d, sin α r / d Rotating p into the -plane about the -ais is equivalent to rotating p onto -ais (b angle α about -ais) (r, r, r ) α d p p Computing the -Rotation α Insert in the rotation matri about -ais: Note that we do not even compute α cos sin sin cos ) ( d r d r d r d r R α α α α α

20 /29/3 Compu;ng the - Rota;on α Determine the rota;on about the - ais: cos α sin α d α r R (α ) and finall M R (-α ) R (-α ) R () R (α ) R (α ) Once computed, M does the comple rotation with a single matri multiplication per verte OpenGL gives us a simple function that computes M for us: glrotatef( ); 2

21 /29/3 Rotation about Arbitrar Ais Hands-On Session Self-Training Tool for Learning 3D Geometrical Transformations (3gtd) Request handout from the instructor 2

22 /29/3 glutwiretorus(inner, outer, sides, rings); 3D GLUT Primitives glutwirecone(radius, height, slices, stacks); GLUT Primitives 22

23 /29/3 glutwiresphere(radius, slices, stacks); GLUT Primitives GLUT Primitives glutwiretetrahedron() glutwiredodecahedron() glutwireoctahedron() glutwireicosahedron() 23

24 /29/3 GLUT Primitives glutwireteapot(sie); Alternative calls are glutsolid... OpenGL Transformations 24

25 /29/3 Basic OpenGL Transformations Translation: gltranslatef(t, t, t ) gltranslatef(t, t, ) à for 2D Rotation: Scaling: glrotatef(angle, r, r, r ) glrotatef(angle,,, ) à for 2D glscalef(s, s, s ) glscalef(s, s, ) à for 2D OpenGL Transformation Matrices Current Transformation Matri (CTM) The CTM is a 4 4 matri Part of the OpenGL state Operations: 25

26 /29/3 Model-View and Projection Matrices Direct Rotation/Translation/Scaling 26

27 /29/3 Composed Transformation Eample Q: What kind of operation is this? Order of Transformations (right to left) 27

28 /29/3 The MODELVIEW Matri glmatrimode (GL_MODELVIEW); modelview mode tells OpenGL that we will be specifing geometric transformations. The command simpl sas that the current matri operations will be applied on the MODELVIEW matri. The other mode is the projection mode, which specifies the matri that is used for projection transformations (i.e., how a scene is projected onto the screen) Transforma;ons Eample MODELVIEW Matri: Before a point is drawn on the displa, it is mul;plied b this matri to get the transformed loca;on void DrawPoint() { glbegin( GL_POINTS ); glverte2f(., 3. ); glend(); } Point loca;on on displa: MV 3 28

29 /29/3 MODELVIEW Matri void Displa() { glmatrimode(gl_modelview); glloadidentit(); // MV DrawPoint(); // point# gltranslatef(4., 2.,, ); // MV DrawPoint(); // point#2 MODELVIEW Matri glrotatef(3.,,, ); // MV DrawPoint(); // point#3 glscalef(2, 2, ); // MV DrawPoint(); // point#4 } // end Displa 29

30 /29/3 Consider This! Suppose ou have a func;on called DrawSquare() that draws a square centered at the origin with sides. Write an OpenGL code fragment that draws the table below. The table top is centered at (,) and has dimensions 2. The table legs have dimensions 24 and the aiach to the underside of the table top at points (- 4,- ) and (4,- ). To get ou started, two lines of code are included below. Your code should call DrawSquare() three ;mes to draw the table top and legs. glmatrimode( GL_MODELVIEW ); glloadidentit(); Consider This! glmatrimode( GL_MODELVIEW ); glloadidentit(); 3

31 /29/3 glpushmatri and glpopmatri Mo;va;ng Eample: void Displa() { gltranslatef( 8, 8, ); // MV } DrawRing(); // 64 DrawPlanet(); // 88 Use DrawUnitCircle to implement Displa Mo;va;ng gl(push,pop)matri void DrawRing() { // MV glrotatef( 45,,, ); glscalef( 8, 2, ); // MV } DrawUnitCircle(); 3

32 /29/3 Mo;va;ng gl(push,pop)matri void DrawPlanet() { // MV glscalef( 4, 4, ); // MV } DrawUnitCircle(); Using gl(push,pop)matri void Displa() { gltranslatef( 8, 8, ); } // MV DrawRing(); DrawPlanet(); void DrawRing() { glpushmatri(); glrotatef( 45,,, ); glscalef( 8, 2, ); DrawUnitCircle(); glpopmatri(); } void DrawPlanet() { glpushmatri(); glscalef( 4, 4, ); DrawUnitCircle(); glpopmatri(); } 32

33 /29/3 Prac;ce gl(push,pop)matri glloadidentit(); MV I glpushmatri(); MV I gltranslatef(,, ); MV T(,) glscalef( 2, 2, ); MV T(,)S(2,2) glpushmatri(); gltranslatef( 2, 2, ); glpushmatri(); glrotatef( 3,,, ); glpopmatri(); glpopmatri(); gltranslatef( 3, 3, ); glpopmatri(); Hands-On Session OpenGL Transformation Applet at applets/opengltransformations.html Request handout from the instructor 33

34 /29/3 Summar Transformations Translation (required direction vector) Scaling (requires fied point, scaling direction) Rotation (requires fied point, vector, angle) Combining transformations Order is important 34