1/50 CS148: Introduction to Computer Graphics and Imaging Transforms CS148: Introduction to Computer Graphics and Imaging Transforms

2/50 © Louie Psihoyos

3/50 Rotate glRotatef(angle,ax,ay,az)

4/50 Translate glTranslatef(tx,ty,tz)

5/50 Scale Uniform Nonuniform glScalef(sx,sy,sz)

6/50 Transformations What? Just functions acting on points (x’,y’,z’) = T(x,y,z) P’ = T(P) Why? Viewing ￭ Window coordinates to framebuffer coordinates ￭ Virtual camera: parallel/perspective projections Modeling ￭ Create objects in convenient coordinates ￭ Multiple instances of a prototype shape ￭ Kinematics of linkages/skeletons - characters/robots

7/50 Scale

8/50 Scale

9/50 Reflection

10/50 Shear

11/50 Linear Transform = Matrices

12/50 Lines Map to Lines→ Linear Transform →

13/50 Non-Linear Transform! Lines Map to Curves

14/50 Parametric Form of a Line

15/50 Linear Transform→ Lines Map to Lines Start with the parametric form of a line Transform all the points on the line Thus, a line transforms into a linear combination of transformed points, which is a line

16/50 Coordinate System An origin o and two unit vectors x, y define a frame of reference

17/50 Points are Defined wrt to the Frame

18/50 Coordinate Frames Can interpret the columns of the matrix as the x and y axes that define the frame

19/50 Rotation Matrix?

20/50 Rotation Matrix Setting the columns of the matrix to the x and y axes creates the transform from that frame

21/50 Rotation Matrix

22/50

23/50 Translate??

24/50 Solution: Homogenous Coordinates

25/50 Vectors

26/50 Composing Transformations

27/50 Rotate, Then Translate R(45) T(1,1) R(45)

28/50 Translate, Then Rotate T(1,1) R(45) T(1,1)

29/50 Order Matters R(45) T(1,1)T(1,1) R(45) ≠

30/50 Rotate 45 about the Center (1,1)?

31/50 Rotate (1,1) T(-1,-1) ⇒ ⇓ T(1,1) R(45) T(-1,-1) ⇒ ⇑ R(45) T(1,1)

32/50 Order of Transformations The rightmost transform is applied first P’ = T(1,1) (R(45)(P)) That is, the rotate is applied before the translate OpenGL glTranslatef( 1.0, 1.0, 0.0 ); glRotatef( 45.0, 0., 0., 1. ); The last gl call is applied first; we’ll see why later

33/50 Advantages of the Matrices Combine a sequence of transforms into a single transform P’ = A ( B ( C ( D ( p ) ) ) ) = ( A B C D ) P = M P Why? Matrix multiplication is associative Advantages ￭ Very inefficient to recompute the matrix entries ￭ Very inefficient to multiply matrix times point multiple times ￭ Compute the matrix M once and then apply that matrix to many points

34/50 Transforming Points vs. Transforming Coordinate Systems

35/50 Transforming the Coordinate System T(1,1) R(45) T(1,1) Transforms are defined wrt the Global/World C.S. R(45) T(1,1) R(45)

36/50 Transforming the Coordinate System T(1,1) T(1,1) R(45) R(45) R(45) T(1,1) Transforms are defined wrt the Local/Object C.S.

37/50 Two Interpretations are Equivalent T(1,1) R(45) T(1,1) R(45) R(45) T(1,1) World: Read right to leftObject: Read left to right

38/50 Two Interpretations are Equivalent R(45) T(1,1) R(45) T(1,1) T(1,1) R(45) World: Read right to leftObject: Read left to right

39/50 Two Interpretations are Equivalent R(45) T(1,1) R(45) T(1,1) T(1,1) R(45) World: Read right to leftObject: Read left to right Object way will be very useful when creating hierarchies

40/50 Hierarchical Transformations

41/50 Skeleton body torso head shoulder larm upperarm lowerarm hand rarm upperarm lowerarm hand hips lleg upperleg lowerleg foot rleg upperleg lowerleg foot

42/50 Skeletons and Linkages 1. Skeleton represents the hierarchy of an assembly of parts - The arm is made of an upper arm and a lower arm - Different parts may have the same shape, i.e. are “geometric instances” 2. Parts connected by joints - “The ankle joint is connected to the knee joint” - Different types of joints move differently e.g. a ball and socket joint, a linear actuator

43/50 How Linkages Work Lower levels of the hierarchy move when the upper levels moves ￭ e.g. moving the left shoulder moves the left arm and the left hand Motion in one sub-tree does not effect the position of a part in another sub-tree ￭ e.g. moving the left hand does not effect the right hand

44/50 Pat Hanrahan, Fall 2011 Skeleton Text translate(0,4,0); torso(); pushmatrix(); translate(0,3,0); shoulder(); pushmatrix(); rotatey(necky); rotatex(neckx); head(); popmatrix(); pushmatrix(); translate(1.5,0,0); rotatex(lshoulderx); upperarm(); pushmatrix(); translate(0,-2,0); rotatex(lelbowx); lowerarm();... popmatrix();... translate(0,4,0); torso(); pushmatrix(); translate(0,3,0); shoulder(); pushmatrix(); rotatey(necky); rotatex(neckx); head(); popmatrix(); pushmatrix(); translate(1.5,0,0); rotatex(lshoulderx); upperarm(); pushmatrix(); translate(0,-2,0); rotatex(lelbowx); lowerarm();... popmatrix();...

45/50 OpenGL

46/50 Current Transformation Matrix OpenGL maintains a current transformation matrix (CTM). All geometry is transformed by the CTM. Transformation commands are concatenated onto the CTM on the right. CTM *= T This causes T to be applied before the old CTM

47/50 OpenGL Matrix Functions glLoadIdentity( ) ￭ Set the transform matrix to the identity matrix glLoadMatrix( matrix M ) ￭ Replace the transform matrix with M glMultMatrix( matrix M ) ￭ Multiplies transform matrix by CTM * M ￭ glRotate, glTranslate, glScale etc. are just wrappers for glMultMatrix

48/50 Graphics Coordinate Frames Object ￭ Raw values as provided by glVertex (ex. teacup centered at origin) World ￭ Object at final location in environment (ex. teacup recentered to be on top of a table) Screen ￭ Object at final screen position

49/50 OpenGL Matrix Functions glMatrixMode( mode ) ￭ Sets which transformation matrix to modify ￭ GL_MODELVIEW: object to world transform ￭ GL_PROJECTION: world to screen transform ￭ CTM = GL_PROJECTION * GL_MODELVIEW Why? May need to transform by one of these matrices for certain calculations like lighting. More on this latter.

50/50 OpenGL Matrix Stack Save and restore transformations glPushMatrix ( ) ￭ Pushes the current matrix onto the top of the matrix stack glPopMatrix ( ) ￭ Pops the matrix off the top of the matrix stack and loads it as the current matrix