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# Modeling Transformations

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Modeling transformations

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### Modeling Transformations

1. 1. MODELING TRANSFORMATION 1
2. 2. Modeling Transformations 2D Transformations 3D Transformations OpenGL Transformation 2
3. 3. 2D-Transformations Basic Transformations Homogeneous coordinate system Composition of transformations 3
4. 4. Translation – 2D Y Y (4,5) (7,5) (7,1) Before Translation x’ = x + dx y’ = y + dy X Translation by (3,-4) d x   x  x′  P =   P′ =   T =    y  y ′ d y  Homogeniou s Form  x ′  1 0 d x   x   y ′ =  0 1 d  *  y  y       1  0 0 1   1        (10,1) X P′ = P + T 4
5. 5. Scaling – 2D Y Y (4,5) Types of Scaling: Differential ( sx != sy ) Uniform ( sx = sy ) (7,5) (2,5/4) X Before Scaling (7/2,5/4) Scaling by (1/2, 1/4) X x′ = s x * x y′ = s y * y S * P = P′ Homogeniou s Form ⇓ ⇓ ⇓  x′   s x  y ′ =  0    1 0    sx 0  0   x   x * sx  *  = y*s  sy   y  y 0 sx 0 0  x  0 *  y     1  1     5
6. 6. te d rota Rotation – 2D r cos φ  v= r sin φ    r cos( φ + θ )  v′ =  r sin ( φ + θ )     x′ = r cos φ cosθ − r sinφ sinθ expand (φ + θ ) ⇒   y ′ = r cos φ sinθ − r sinφ cosθ x = r cos φ x′ = x cos θ − y sinθ but ⇒ y = r sinθ y ′ = x sinθ + y cosθ or l i na ig
7. 7. Rotation – 2D Y Y Before Rotation Rotation of 45 deg. w.r.t. origin (4.9,7.8) (2.1,4.9) (5,2) (9,2) X X x * cosθ − y * sin θ = x′ x * sin θ + y * cosθ = y′ R *P= P′  cosθ − sin θ   x   x * cosθ − y * sin θ   sin θ cosθ  *  y  =  x * sin θ + y * cosθ        Homogenious Form  x ′   cosθ  y ′ =  sin θ    1  0    − sin θ cosθ 0 0  x  0 *  y     1  1 7   
8. 8. Mirror Reflection Y Y (1,1) (-1,1) (1,1) X X (1,-1) Reflection about X - axis x′ = x y′ = − y Reflection about Y - axis x′ = − x y ′ = y 1 0 0 M x = 0 − 1 0    0 0 1     − 1 0 0 M y =  0 1 0    0 0 1   8
9. 9. Shearing Transformation 1 a 0 SH x = 0 1 0   0 0 1    unit cube Sheared in X direction 1 0 0  SH y = b 1 0   0 0 1    Sheared in Y direction SH xy 1 a 0 = b 1 0   0 0 1    Sheared in both X and Y direction 9
10. 10. Inverse 2D - Transformations -1 (dx,dy) Translaiton : T Rotation Sclaing -1 (θ ) : R : S = R(-θ ) -1 (sx,sy) Mirror Ref : M M -1 x -1 y = T(-dx,-dy) = S ( 1 sx , 1 sy ) = Mx = My 10
11. 11. Homogeneous Co-ordinates  Translation, scaling and rotation are expressed (non-homogeneously) as: – translation: P′ = P + T – Scale: P′ = S · P – Rotate: P′ = R · P  Composition is difficult to express, since translation not expressed as a matrix multiplication  Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 × 3 matrices  W is 1 for affine transformations in graphics 11
12. 12. Homogeneous Co-ordinates  P2d is a projection of Ph onto the w = 1 plane  So an infinite number of points correspond to : they constitute the whole line (tx, ty, tw) w Ph(x,y,w) w=1 P2d(x,y,1) y x 12
13. 13. Classification of Transformations 1. Rigid-body Transformation  Preserves parallelism of lines  Preserves angle and length  e.g. any sequence of R(θ) and T(dx,dy) 2. Affine Transformation  Preserves parallelism of lines  Doesn’t preserve angle and length  e.g. any sequence of R(θ), S(sx,sy) and T(dx,dy) unit cube 45 deg rotaton Scale in X not in Y 13
14. 14. Properties of rigid-body transformation The following Matrix is Orthogonal if the upper left 2X2 matrix has the following properties 1. A) Each row are unit vector. sqrt(r11* r11 + r12* r12) = 1 B) Each column are unit vector. sqrt(c11* c11 + c12* c12) = 1 2. A) Rows will be perpendicular to each other (r11 , r12 ) . ( r21 , r22) = 0  r11 r  21 0  cosθ  sin θ   0  r12 r22 0 − sin θ cos θ 0 tx  ty   1  0 0  1  B) Columns will be perpendicular to each other (c11 , c12 ) . (c21 ,c22) = 0 e.g. Rotation matrix is orthogonal • Orthogonal Transformation ⇒ Rigid-Body Transformation • For any orthogonal matrix B ⇒ B -1 = B T 14
15. 15. Commutativity of Transformation Matrices • In general matrix multiplication is not commutative • For the following special cases commutativity holds i.e. M1.M2 = M2.M1 M1 M2 Translate Translate Scale Scale Rotate Rotate Uniform Scale Rotate • Some non-commutative Compositions:  Non-uniform scale, Rotate  Translate, Scale  Rotate, Translate Original Transitional Final 15
16. 16. Associativity of Matirx Multiplication Create new affine transformations by multiplying sequences of the above basic transformations. q = CBAp q = ( (CB) A) p = (C (B A))p = C (B (Ap) ) etc. matrix multiplication is associative. To transform just a point, better to do q = C(B(Ap)) But to transform many points, best to do M = CBA then do q = Mp for any point p to be rendered. For geometric pipeline transformation, define M and set it up with the model-view matrix and apply it to any vertex subsequently defined to its setting. 16
17. 17. Rotation of θ about P(h,k): Rθ,P Step 1: Translate P(h,k) to origin Step 2: Rotate θ w.r.t to origin Q3(x’+h, y’ +k) Step 3: Translate (0,0) to P(h,k0) R θ,P = T(h ,k) * R θ * T(-h ,-k) P3(h,k) Q(x,y) P(h,k) Q1(x’,y’) P1 (0,0) Q2(x’,y’) P2 (0,0) 17
18. 18. Scaling w.r.t. P(h,k): Ssx,sy,p Step 1: Translate P(h,k) to origin Step 2: Scale S(sx,sy) w.r.t origin (7,2) Step 3: Translate (0,0) to P(h,k) S sx,sy,P = T(h ,k) * S(s x ,s y )* T(-h ,-k) (4,3) (1,1) (1,1) T(1,1) (4,2) (6,1) (4,1) S 3/2,1/2,(1,1) (0,0) (4,0) T(-1,-1) (7,1) (0,0) (6,0) S(3/2,1/2) 18
19. 19. Reflection about line L, ML Y Step 1: Translate (0,b) to origin Step 2: Rotate -θ degrees Step 3: Mirror reflect about X-axis Step 4: Rotate θ degrees (0,b) t O X Step 5: Translate origin to (0,b) M L = T(0 ,b) * R(θ) * M x * R(-θ) * T(0 ,-b) 19
20. 20. Problems to be solved: Schaum’s outline series: Problems:  4.1  4.2  4.3, 4.4, 4.5 => Rθ,P  4.6, 4.7, 4.8 => S sx,sy,,P  4.9, 4.10, 4.11, 4.21 => ML  4.12 => Shearing  Pg-281(1.24), Pg-320(5.19) => Circular view-port 20
21. 21. 3D Transformations Basics of 3D geometry Basic 3D Transformations Composite Transformations 21
22. 22. Orientation Thumb points to +ve Z-axis Fingers show +ve rotation from X to Y axis Y Y Z (larger z are away from viewer) X X Z (out of page) Right-handed orentation Left-handed orentation 22
23. 23. Vectors in 3D Have length and direction V = [xv , yv , zv] Length is given by the Euclidean Norm ||V|| = √( xv2 + yv2 + zv2 ) Dot Product V • U = [xv, yv, zv]•[xu, yu, zu] = xv*xu + yv*yu + zv*zu = ||V|| || U|| cos ß Cross Product V×U = [yv*zu - zv yu , -xv*zu + zv*xu , xv*yu – yv*xu ] = η ||V|| || U|| sin ß V × U = - ( U x V) z K J+c b aI+ V= (xv,yv,zv) y x 23
24. 24. 3D Equation of Curve & Line  Parametric equations of Curve & Line  Curve  Line x = f (t) C : y = g(t ) z = h( t ) z C a≤t ≤b y x V = P0 P1 = P − P0 1 x = x0 + ( x1 − x0 ) × t t =1 L : y = y0 + ( y1 − y0 ) × t 0 ≤ t ≤ 1 z = z0 + ( z1 − z0 ) × t L = P0 + t ( P − P0 ) = P0 + tV 1 t t= <0 0 1 t< 0< V P0(x0,y0,z0) t> 1 P1(x1,y1,z1) 24
25. 25. 3D Equation of Surface & Plane  Parametric equations of Surface & Plane  Surface x = f ( s, t ) S : y = g ( s, t ) z = h( s , t ) a≤s≤b c≤t ≤d  Plane : with Normal, N Ax + By + Cz + D = 0 ˆ ˆ N = Ai + Bˆ + Ck j N P0 25
26. 26. 3D Plane  Ways of defining a plane 1. 3 points P0, P1, P2 on the plane 2. Plane Normal N & P0 on plane 3. Plane Normal N & a vector V on the plane N Plane Passing through P0, P1, P2 ˆ ˆ N = P0 P × P0 P2 = Ai + Bˆ + Ck j 1 P0 if P(x, y, z) is on the plane N • P0 P = 0 [ P1 P2 V ] ˆ ˆ ˆ ˆ ⇒ ( Ai + Bˆ + Ck ) • ( x − x0 )i + ( y − y0 ) ˆ + ( z − z0 )k = 0 j j ⇒ Ax + By + Cz + D = 0 where D = −( Ax0 + By0 + Cz0 ) 26
27. 27. Affine Transformation  Transformation – is a function that takes a point (or vector) and maps that point (or vector) into another point (or vector).  A coordinate transformation of the form: x’ = axx x + axy y + axz z + bx , y’ = ayx x + ayy y + ayz z + by , z’ = azx x + azy y + azz z + bz ,  x'   a xx     y '   a yx  z'  =  a    zx  w  0    a xy a xz a yy a zy a yz a zz 0 0 bx  x    b y  y  bz  z    1  1    is called a 3D affine transformation.  The 4th row for affine transformation is always [0 0 0 1].  Properties of affine transformation: – translation, scaling, shearing, rotation (or any combination of them) are examples affine transformations. – Lines and planes are preserved. – parallelism of lines and planes are also preserved, but not angles and length. 27
28. 28. Translation – 3D x′ = x + d x y′ = y + d y z′ = z + d z 1 0  0  0 0 0 dx   x  x + dx  1 0 d y   y  y + d y  *  =   0 1 dz   z   z + dz       0 0 1  1   1  ⇓ ⇓ T (d x , d y , d z ) * P = ⇓ P′ 28
29. 29. Scaling – 3D x′ = s x * x y′ = s y * y Original scale Y axis S (sx , s y , sz ) ⇓ scale all axes z′ = sz * z sx 0  0  0 * P = ⇓ 0 0 sy 0 0 sz 0 0 P′ ⇓ 0  x   x * s x  0  y   y * s y  *  =   0  z   z * s z       29 1  1   1 
30. 30. Rotation – 3D For 3D-Rotation 2 parameters are needed Rotation about z-axis:  Angle of rotation  Axis of rotation Rθ ,k P′ ⇓ cosθ  sin θ   0   0 * P = ⇓ ⇓ − sin θ cos θ 0 0 0 0  x   x * cos θ − y * sin θ  0 0  y   x * sin θ + y * cosθ  *  =    1 0  z   z      0 1  1   1  30
31. 31. Rotation about Y-axis & X-axis About yaxis Rθ , j * P = P′ ⇓ ⇓ ⇓  cos θ  0  − sin θ   0 About x-axis 0 sin θ 1 0 0 cosθ 0 0 0  x   x * cosθ + z * sin θ   0  y   y *  =   0  z  − x * sin θ + z * cosθ       1  1   1  * P = P′ ⇓ 0 1 0 cos θ  0 sin θ  0 0 Rθ , i ⇓ ⇓ 0 − sin θ cosθ 0 0  x   x  0  y   y * cosθ − z * sin θ  *  =   0  z   y * sin θ + z * cosθ       1  1   1  31
32. 32. Shear along Z-axis y x z SH xy ( shx , sh y ) * P = ⇓ 1 0  0  0 ⇓ 0 shx 1 sh y 0 1 0 0 P′ ⇓ 0  x   x + z * shx    y   y + z * sh  0    y * =  0  z   z      1  1   1  32
33. 33. Object Transformation Line: Can be transformed by transforming the end points Plane:(described by 3-points) Can be transformed by transforming the 3-points Plane:(described by a point and Normal) Point is transformed as usual. Special treatment is needed for transforming Normal 33
34. 34. Composite Transformations – 3D Some of the composite transformations to be studied are: AV,N = aligning a vector V with a vector N Rθ,L = rotation about an axis L( V, P ) Ssx,sy,P= scaling w.r.t. point P 34
35. 35. AV : aligning vector V with k Step 1 : Rotate about x - axis by θ b sin θ =  λ  λ = b2 + c2  c cos θ =  λ z ( 0, 0, λ) ( 0, b,c b ( a, 0, λ) ) cλ θ λ |V| θ V = aI + bJ + cK |V| k b y a Av = Rθ,i 35 x
36. 36. AV : aligning vector V with k Step 1 : Rotate about x - axis by θ b sin θ =  λ  λ = b2 + c2  c cos θ =  λ z ( 0, 0, |V|) z a ( 0, b,c b ( a, 0, λ) −a  |V|  2 2 2  | V |= a + b + c λ  cos(−φ ) = | V |  λ Step 2 : Rotate V about y - axis by - φ ) c |V| |V| P( a, b, c) ϕ sin( −φ ) = Av = R-ϕ,j * Rθ,i k b y a 36 x
37. 37. AV : aligning vector V with k  AV-1 = AVT  AV,N = AN-1 * AV  0 AV =  a V  0 λ V - ab λV c λ b V - ac λV −b λ c V 0 0 0  0 0  1 37
38. 38. Rθ,L : rotation about an axis L Let the axis L be represented by vector V and z passing through point P 1. Translate P to the origin 2. Align V with vector k 3. Rotate θ° about k 4. Reverse step 2 5. Reverse step 1 x L P Q V k Rθ,L = T-P-1 * AV-1 * Rθ,k * AV * T-P θ Q' y 38
39. 39. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane z N P x y 39
40. 40. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin z N P x y MN,P = T-P 40
41. 41. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin z N 2. Align N with vector k P x y MN,P = AN * T-P 41
42. 42. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k 3. Reflect w.r.t xy-plane z N P x y MN,P = S1,1,-1 * AN * T-P 42
43. 43. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin z 2. Align N with vector k 3. Reflect w.r.t xy-plane x 4. Reverse step 2 MN,P = AN-1 * S1,1,-1 * AN * T-P y 43
44. 44. MN,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane 1. Translate P to the origin 2. Align N with vector k 3. Reflect w.r.t xy-plane x 4. Reverse step 2 5. Reverse step 1 z N P MN,P = T-P-1 * AN-1 * S1,1,-1 * AN * T-P y 44
45. 45. Further Composition  Translate points in fig. 1 into points in fig 2 such that: – P3 is moved to yz plane – P2 is on z-axis – P1 is at Origin  The Composite Transform must have – Translation of P1 to Origin ⇒ T – Some Combination of Rotations ⇒ R y y P3 P2 P 1 x z Fig. 1 y P3′ T R P2′ P′ 1 z P3′′ x z P2′′ P′′ 1 Fig. 2 x 45
46. 46. Finding R Let R be  r11 r12 r13   Rx .x Rx . y Rx .z  R = r21 r22 r23  =  R y .x R y . y R y .z       r31 r32 r33   Rz .x Rz . y Rz .z      R is Rigid - body Transform i) Rx , R y , Rz are unit vectors ii) Rx , R y , Rz are perpendicu lar to each other Note : Rx .x ⇒ x component of vextor Rx 46
47. 47. Finding Rz R aligns P1′P2′ along z - axis ⇒ R⋅ P1′P2′ P1′P2′ P′ 3 ˆ =k ˆ = P1′P2′ ⇒ R ⋅k P1′P2′ T  Rx . x  ⇒  Rx . y  Rx .z  y R y .x Ry . y R y .z Rz [ R −1 =R T P′ 1 ] x z R Rz . x  0  P ′P ′  R z . y  0  = 1 2   ′ ′ Rz .z  1 P1P2    Rz . x   R . y  = P1′P2′ = R ⇒ z  z  Rz .z  P1′P2′   P2′ y ′ P′ 3 ˆ k z ′ P2′ ′ P′ 1 x 47
48. 48. Finding Rx R aligns P1′P3′ × P1′P2′ along x - axis ⇒ R⋅ P1′P3′ × P1′P2′ P1′P3′ × P1′P2′ ˆ ⇒ R ⋅i = −1  Rx . x  ⇒  Rx . y  Rx .z  P′ 3 ˆ =i P′ 1 P1′P3′ × P1′P2′ P2′ x Rx R R z . x  1  P ′P ′ × P ′P ′  R z . y  0  = 1 3 1 2   ′ ′ ′ ′ Rz .z  0 P1P3 × P1P2    Rx . x   R . y  = P1′P3′ × P1′P2′ = R ⇒ x  x  Rx .z  P1′P3′ × P1′P2′   Rz z P1′P3′ × P1′P2′ R y .x Ry . y R y .z y y ′ P′ 3 ˆ k z ′ P2′ ′ P′ 1 ˆ i x 48
49. 49. Finding Ry y R aligns R z × R x along y - axis ( ) P′ 3 ⇒ R ⋅ Rz × Rx = ˆ j P′ 1 ⇒ R ⋅ ˆ = Rz × Rx j R y .x Ry . y R y .z Ry x R Rz . x  0   Rz . y  1 = R z × R x   R z . z  0     R y .x    ⇒  Ry . y  = Rz × Rx = Ry  R y .z    P2′ Rx z −1  Rx . x  ⇒  Rx . y  Rx .z  Rz y ′ P′ 3 ˆ k z ′ P2′ ˆ j ′ P′ 1 ˆ i x 49
50. 50. Problems to be solved: Schaum’s outline series: Problems:  6.1  6.2, 6.5, 6.9, 6.10, 6.11, 6.12 ⇒ Av  6.3, 6.4 ⇒ Rθ,L  6.6, 6.7, 6.8 ⇒ MN,P 50
51. 51. Transformations in OpenGL OpenGL transformation commands Transformation Order Hierarchical Modeling 51
52. 52. Transformations in OpenGL  OpenGL uses 3 stacks to maintain transformation matrices: – Model & View transformation matrix stack – Projection matrix stack – Texture matrix stack  You can load, push and pop the stack  The top most matrix from each stack is applied to all graphics primitive until it is changed Graphics Primitives (P) M Model-View Matrix Stack N Projection Matrix Stack Output N•M•P 52
53. 53. General Transformation Commands  Specify current matrix (stack) : – void glMatrixMode(GLenum mode) I B A • Mode : GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE glLoadMatrix(M) glLo a dIde ntity C B A M B A  Initialize current matrix. – void glLoadIdentity(void) • Sets the current matrix to 4X4 identity matirx – void glLoadMatrix{f|d}(cost TYPE *M) • Sets the current matrix to 4X4 matrix specified by M Note: current matrix ⇒ Top most matrix of the current matrix stack 53
54. 54. General Transformation Commands Concatenate Current Matrix: – void glMultMatrix(const TYPE *M) • Multiplies current matrix C, by M. i.e. C = C*M – Caveat: OpenGL matrices are stored in column major order.  m1 m  2  m3  m4 m5 m6 m7 m8 m9 m10 m11 m12 m13  m14   m15   m16  – Best use utility function glTranslate, glRotate, glScale for common transformation tasks. 54
55. 55. Transformations and OpenGL®  Each time an OpenGL transformation M is called the current MODELVIEW matrix C is altered: v′ = Cv v′ = CMv glTranslatef(1.5, 0.0, 0.0); glRotatef(45.0, 0.0, 0.0, 1.0); v′ = CTRv Note: v is any vertex placed in rendering pipeline v’ is the transformed vertex from v. 55
56. 56. Sample Instance Transformation glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(...); glRotatef(...); glScalef(...); gluCylinder(...); 56
57. 57. Thinking About Transformations  There is a World Coordinate System where:  All objects are defined  Transformations are in World Coordinate space Two Different Views As a Global System  Objects moves but coordinates stay the same  Think of transformation in reverse order as they appear in code As a Local System  Objects moves and coordinates move with it  Think of transformation in same order as they appear in code 57
58. 58. Order of Transformation T•R Global View  Rotate Object  Then Translate glLoadIdentity(); Local View glMultiMatrixf( T);  Translate Object glMultiMatrixf( R);  Then Rotate draw_ the_ object( v); v’ = ITRv Effect is same, but perception is different 58
59. 59. Order of Transformation R•T Global View  Translate Object  Then Rotate glLoadIdentity(); Local View glMultiMatrixf( R);  Rotate Object glMultiMatrixf( T);  Then Translate draw_ the_ object( v); v’ = ITRv Effect is same, but perception is different 59
60. 60. Hierarchical Modeling  Many graphical objects are structured  Exploit structure for – Efficient rendering – Concise specification of model parameters – Physical realism  Often we need several instances of an object – Wheels of a car – Arms or legs of a figure – Chess pieces     Encapsulate basic object in a function Object instances are created in “standard” form Apply transformations to different instances Typical order: scaling, rotation, translation 60
61. 61. OpenGL & Hierarchical Model Some of the OpenGL functions helpful for hierarchical modeling are: – void glPushMatrix(void); – void glPoipMatrix(void); – void glGetFloatv(GL_MODELVIEW_MATRIX, *m); s glPu C B A rix Mat h glP ush Ma trix C C B A B A C B A glGetFloatv m C 61
62. 62. Scene Graph  A scene graph is a hierarchical representation of a scene  We will use trees for representing hierarchical objects such that: – Nodes represent parts of an object – Topology is maintained using parent-child relationship – Edges represent transformations that applies to a part and all the subparts connected to that part Scene typedef struct treenode { GLfloat m[16]; // Transformation Sun Star X void (*f) ( ); // Draw function struct treenode *sibling; Earth Venus Saturn struct treenode *child; } treenode; Moon Ring 62
63. 63. Example - Torso  Initializing transformation matrix for node treenode torso, head, ...; /* in init function */ glLoadIdentity(); glRotatef(...); glGetFloatv(GL_MODELVIEW_MATRIX, torso.m);  Initializing pointers torso.f = drawTorso; torso.sibling = NULL; torso.child = &head; 63
64. 64. Generic Traversal To render the hierarchy: – Traverse the scene graph depth-first – Going down an edge: • push the top matrix onto the stack • apply the edge's transformation(s) – At each node, render with the top matrix – Going up an edge: • pop the top matrix off the stack 64
65. 65. Generic Traversal : Torso  Recursive definition void traverse (treenode *root) { if (root == NULL) return; glPushMatrix(); glMultMatrixf(root->m); root->f(); if (root->child != NULL) traverse(root->child); glPopMatrix(); if (root->sibling != NULL) traverse(root->sibling); }  C is really not the right language for this !! 65
66. 66. Viewing Transformation 66
67. 67. Viewing Pipeline Revisited Graphics Primitives Modeling Transform Po Object Coordinates Pw Viewing Transform Pe World Coordinates Eye Coordinates ye yw xe yo -ze pe po pw xo xw zo zw 67
68. 68. Viewing Transformation in OpenGL  To setup the modelview matrix, OpenGL provides the following function: gluLookAt( eyex, eyey, eyez, centerx, centery, centerz, upx, upy, upz ) up (upx, upy, upz) eye (eyex, eyey, eyez) z y center (centerx, centery, centerz) x 68
69. 69. Implementation We want to construct an Orthogonal Frame such that, (1) its origin is the point eye (2) its -z basis vector points towards the point center (3) the up vector projects to the up direction (+ve y-axis) Let C (for camera) denote this frame. Clearly, C.O = eye  v = normalize( center − eye )   C.ez = −v    C.ex = normalize( v × up )    C.e y = ( C.ez × C.ex ) C.e y up (upx, upy, upz) v center C.O (eye) C.e C.e x 69 z
70. 70. Thank You 70