Modeling Transformations

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Modeling transformations

Modeling transformations

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  • 1. MODELING TRANSFORMATION 1
  • 2. Modeling Transformations 2D Transformations 3D Transformations OpenGL Transformation 2
  • 3. 2D-Transformations Basic Transformations Homogeneous coordinate system Composition of transformations 3
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 3D Transformations Basics of 3D geometry Basic 3D Transformations Composite Transformations 21
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Transformations in OpenGL OpenGL transformation commands Transformation Order Hierarchical Modeling 51
  • 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. 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. 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. 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. Sample Instance Transformation glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(...); glRotatef(...); glScalef(...); gluCylinder(...); 56
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. Viewing Transformation 66
  • 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. 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. 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. Thank You 70