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

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

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

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

Rotation – 2D rotated original

6 Rotation – 2D

7 Mirror Reflection

8 Shearing Transformation

9 Inverse 2D - Transformations

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

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

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

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 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 [ ].  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 Translation – 3D

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

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

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

31 Shear along Z-axis y x z

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

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

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

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 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 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 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 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 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 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 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 Finding R

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

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

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

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

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

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 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 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 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 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 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 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 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 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 Viewing Transformation

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 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 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 Thank You