1. 1 Modeling Transformations  2D Transformations  3D Transformations  OpenGL Transformation

2. 2 2D-Transformations  Basic Transformations  Homogeneous coordinate system  Composition of transformations

3. 3 Translation – 2D x’ = x + d x y’ = y + d y

4. 4 Scaling – 2D Types of Scaling:  Differential ( s x != s y )  Uniform ( s x = s y )

5. Rotation – 2D rotated original

6. 6 Rotation – 2D

7. 7 Mirror Reflection

8. 8 Shearing Transformation

9. 9 Inverse 2D - Transformations

10. 10 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  matrices  W is 1 for affine transformations in graphics

11. 11 Homogeneous Co-ordinates  P 2d is a projection of P h onto the w = 1 plane  So an infinite number of points correspond to : they constitute the whole line (tx, ty, tw) x y w P h (x,y,w) P 2d (x,y,1) w=1

12. 12 Classification of Transformations 1.Rigid-body Transformation Preserves parallelism of lines Preserves angle and length e.g. any sequence of R(  ) and T(dx,dy) 2.Affine Transformation Preserves parallelism of lines Doesn’t preserve angle and length e.g. any sequence of R(  ), S(sx,sy) and T(dx,dy)

13. 13 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(r 11 * r 11 + r 12 * r 12 ) = 1 B) Each column are unit vector. sqrt(c 11 * c 11 + c 12 * c 12 ) = 1 2.A) Rows will be perpendicular to each other (r 11, r 12 ). ( r 21, r 22 ) = 0 B) Columns will be perpendicular to each other (c 11, c 12 ). (c 21,c 22 ) = 0 e.g. Rotation matrix is orthogonal Orthogonal Transformation  Rigid-Body Transformation For any orthogonal matrix B  B -1 = B T

14. 14 Commutativity of Transformation Matrices In general matrix multiplication is not commutative For the following special cases commutativity holds i.e. M 1.M 2 = M 2.M 1 M1M1 M2M2 Translate Scale Rotate Uniform ScaleRotate Some non-commutative Compositions:  Non-uniform scale, Rotate  Translate, Scale  Rotate, Translate Original Transitional Final

15. 15 Associativity of Matirx Multiplication Create new affine transformations by multiplying sequences of the above basic transformations. q = CBA p q = ( ( CB ) A ) p = ( C ( B A ))p = C ( B ( A p) ) etc. matrix multiplication is associative. But to transform many points, best to do M = CBA then do q = M p for any point p to be rendered. To transform just a point, better to do q = C ( B ( A p)) 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. 16 Rotation of  about P(h,k): R ,P R ,P = Q(x,y) P(h,k) Step 1: Translate P(h,k) to origin T(-h,-k) Q 1 (x ’,y ’ ) P 1 (0,0) Step 2: Rotate  w.r.t to origin R*R* Q 2 (x ’,y ’ ) P 2 (0,0) Step 3: Translate (0,0) to P(h,k0) T(h,k) * P 3 (h,k) Q 3 (x ’ +h, y ’ +k)

17. 17 Scaling w.r.t. P(h,k): S sx,sy,p T(-h,-k)S(s x,s y )*T(h,k) * S sx,sy,P = (4,3) (1,1)(4,1) S 3/2,1/2,(1,1) Step 1: Translate P(h,k) to origin (4,2) (0,0)(4,0) T(-1,-1) Step 2: Scale S(s x,s y ) w.r.t origin (6,1) (6,0)(0,0) S(3/2,1/2) (7,1) Step 3: Translate (0,0) to P(h,k) (7,2) (1,1) T(1,1)

18. 18 Reflection about line L, M L Step 1: Translate (0,b) to origin T(0,-b)M L = Step 2: Rotate -  degrees Step 3: Mirror reflect about X-axis R(-  ) * T(0,b) * Step 4: Rotate  degrees Step 5: Translate origin to (0,b) M x * R(  ) *

19. 19 Problems to be solved: Schaum’s outline series: Problems:  4.1  4.2  4.3, 4.4, 4.5 => R ,P  4.6, 4.7, 4.8 => S sx,sy,,P  4.9, 4.10, 4.11, 4.21 => M L  4.12 => Shearing  Pg-281(1.24), Pg-320(5.19) => Circular view-port

20. 20 3D Transformations  Basics of 3D geometry  Basic 3D Transformations  Composite Transformations

21. 21 Orientation  Thumb points to +ve Z-axis  Fingers show +ve rotation from X to Y axis

22. 22 Vectors in 3D Have length and direction V = [x v, y v, z v ] Length is given by the Euclidean Norm ||V|| =  ( x v 2 + y v 2 + z v 2 ) Dot Product V U = [x v, y v, z v ][x u, y u, z u ] = x v *x u + y v *y u + z v *z u = ||V|| || U|| cos ß Cross Product V  U = [y v *z u - z v y u, -x v *z u + z v *x u, x v *y u – y v *x u ] =  ||V|| || U|| sin ß V  U = - ( U x V) (xv,yv,zv) V=aI+bJ+cK x y z

23. 23 3D Equation of Curve & Line  Parametric equations of Curve & Line  Curve  Line P 0 (x 0,y 0,z 0 ) P 1 (x 1,y 1,z 1 ) t > 1 V t < 0 t =1 t = 0 0 < t < 1 x y z C

24. 24 3D Equation of Surface & Plane  Parametric equations of Surface & Plane  Surface  Plane : with Normal, N P0P0 N

25. 25 3D Plane  Ways of defining a plane 1. 3 points P 0, P 1, P 2 on the plane 2. Plane Normal N & P 0 on plane 3. Plane Normal N & a vector V on the plane Plane Passing through P 0, P 1, P 2 P0P0 P1P1 P2P2 N V

26. 26 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’ = a xx x + a xy y + a xz z + b x, y’ = a yx x + a yy y + a yz z + b y, z’ = a zx x + a zy y + a zz z + b z, is called a 3D affine transformation.  The 4 th 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. 27 Translation – 3D

28. 28 Scaling – 3D Original scale all axes scale Y axis

29. 29 Rotation – 3D For 3D-Rotation 2 parameters are needed  Angle of rotation  Axis of rotation Rotation about z-axis:

30. 30 Rotation about Y-axis & X-axis About x-axis About y-axis

31. 31 Shear along Z-axis y x z

32. 32 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. 33 Composite Transformations – 3D Some of the composite transformations to be studied are:  A V,N = aligning a vector V with a vector N  R ,L = rotation about an axis L( V, P )  S sx,sy,P = scaling w.r.t. point P

34. 34 A V : aligning vector V with k A v =R ,i

35. 35 A V : aligning vector V with k A v =R ,i R - ,j *

36. 36 A V : aligning vector V with k  A V -1 = A V T  A V,N = A N -1 * A V

37. 37 R ,L : rotation about an axis L Let the axis L be represented by vector V and passing through point P 1. Translate P to the origin 2. Align V with vector k 3. Rotate  about k 4. Reverse step 2 5. Reverse step 1 R ,L =T -P A V *R ,k * A V -1 *T -P -1 *

38. 38 M N,P : Mirror reflection Let the plane be represented by plane normal N and a point P in that plane x y z N P

39. 39 M N,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 M N,P =T -P x y z N P

40. 40 M N,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 M N,P =T -P A N * N P x y z

41. 41 M N,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 M N,P =T -P A N *S 1,1,-1 * N P x y z

42. 42 x y z M N,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 4. Reverse step 2 M N,P =T -P A N *S 1,1,-1 *A N -1 *

43. 43 M N,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 4. Reverse step 2 5. Reverse step 1 M N,P =T -P A N *S 1,1,-1 *A N -1 *T -P -1 * x y z N P

44. 44 Further Composition  The Composite Transform must have –Translation of P 1 to Origin  T z x y T –Some Combination of Rotations  R R x y zz x y Fig. 1 Fig. 2  Translate points in fig. 1 into points in fig 2 such that: –P 3 is moved to yz plane –P 2 is on z-axis –P 1 is at Origin

45. 45 Finding R

46. 46 Finding R z R z x y x y z RzRz

47. 47 R x y z z x y Finding R x RxRx RzRz

48. 48 Finding R y R x y z z x y RxRx RzRz RyRy

49. 49 Problems to be solved: Schaum’s outline series: Problems:  6.1  6.2, 6.5, 6.9, 6.10, 6.11, 6.12  A v  6.3, 6.4  R ,L  6.6, 6.7, 6.8  M N,P

50. 50 Transformations in OpenGL  OpenGL transformation commands  Transformation Order  Hierarchical Modeling

51. 51 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 M N Model-View Matrix Stack Projection Matrix Stack Graphics Primitives (P) Output NMP

52. 52 General Transformation Commands  Specify current matrix (stack) : –void glMatrixMode(GLenum mode) Mode : GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE  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 A B C A B I A B M glLoadMatrix(M) glLoadIdentity

53. 53 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. –Best use utility function glTranslate, glRotate, glScale for common transformation tasks.

54. 54 Transformations and OpenGL ®  Each time an OpenGL transformation M is called the current MODELVIEW matrix C is altered: glTranslatef(1.5, 0.0, 0.0); glRotatef(45.0, 0.0, 0.0, 1.0); Note: v is any vertex placed in rendering pipeline v’ is the transformed vertex from v.

55. 55 Sample Instance Transformation glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(...); glRotatef(...); glScalef(...); gluCylinder(...);

56. 56 Thinking About Transformations 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  There is a World Coordinate System where:  All objects are defined  Transformations are in World Coordinate space Two Different Views

57. 57 Local View  Translate Object  Then Rotate Order of Transformation TR glLoadIdentity(); glMultiMatrixf( T); glMultiMatrixf( R); draw_ the_ object( v); v’ = ITRv Global View  Rotate Object  Then Translate Effect is same, but perception is different

58. 58 Order of Transformation RT glLoadIdentity(); glMultiMatrixf( R); glMultiMatrixf( T); draw_ the_ object( v); v’ = ITRv Local View  Rotate Object  Then Translate Global View  Translate Object  Then Rotate Effect is same, but perception is different

59. 59 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. 60 OpenGL & Hierarchical Model A B C C glPushMatrix –void glPushMatrix(void); A B glPushMatrix –void glPoipMatrix(void); A B C C m glGetFloatv –void glGetFloatv(GL_MODELVIEW_MATRIX, *m); A B C  Some of the OpenGL functions helpful for hierarchical modeling are:

61. 61 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 typedef struct treenode { GLfloat m[16];// Transformation void (*f) ( );// Draw function struct treenode *sibling; struct treenode *child; } treenode; Scene SunStar X EarthVenusSaturn MoonRing

62. 62 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. 63 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. 64 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. 65 Viewing Transformation

66. 66 Viewing Pipeline Revisited xwxw ywyw zwzw pwpw Modeling Transform World Coordinate s PwPw yoyo xoxo zozo popo Graphics Primitives PoPo Object Coordinates yeye xexe -z e pepe Viewing Transform PePe Eye Coordinate s

67. 67 Viewing Transformation in OpenGL  To setup the modelview matrix, OpenGL provides the following function: gluLookAt(eyex, eyey, eyez, centerx, centery, centerz, upx, upy, upz )

68. 68 Implementation Orthogonal Frame We want to construct an Orthogonal Frame such that, (1)point eye (1) its origin is the point eye (2)point center (2) its -z basis vector points towards the point center (3)up vector (3) the up vector projects to the up direction (+ve y-axis) C Let C (for camera) denote this frame. Clearly,

69. 69 Thank You