1. Transformations
고려대학교 컴퓨터 그래픽스 연구실
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2. Contents Affine transformations
rotation, translation, and scaling
Transformations in homogeneous coordinates
Concatenation of transformations
rotation about a fixed point
general rotation
instance transformation
rotation about an arbitrary axis
OpenGL transformation matrices
Smooth rotation with a virtual trackball
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3. transformation function
Transformations
Take a point (or vector) and map that point (or vector) into another point (or vector) P Q u v T R
4D column matrices
homogeneous
coordinate
transformation function
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4. Affine Transformations (1/2)
Linearity – linear function
Linear transformation
transform the representation of a point (or vector) into another representation of a point (or vector)
44 matrix
vector
point
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5. Affine Transformations (2/2)
Linear transformation (cont’)
preserve lines – transform a line into another line
 only transform the endpoints of a line segment
Most transformations in CG are affine
rotation, translation, scaling, and shear
homogeneous
coordinate
affine transformation
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6. Translation
Operation that displace points by a fixed distance in a given direction
displacement vector d
(a) object in
original position
(b) object translated
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7. Rotation (1/2)
Simple example of 2D rotation
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8. Rotation (1/2)
Simple example of 2D rotation
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9. Rotation (1/2)
Simple example of 2D rotation
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10. (a) rotation about a fixed point
Needs
fixed point – a point is unchanged by the rotation
rotation angle – positive rotation (counterclockwise in right hand system)
rotation axis in 3D – values on axis are unchanged by the rotation
(a) rotation about a fixed point
(b) 3D rotation
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11. Rigid-Body Transformations
Rotation and translation
No combination of rotations and translations can alter the shape of object
 alter only the object’s location and orientation
affine transformations, but non-rigid body transformations
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12. Scaling (1/2) Make an object bigger or smaller
uniform – scaling in all directions
Affine non-rigid body transformation
affine transformation: translation, rotation, scaling, shear
nonuniform
uniform
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13. Scaling (2/2) Needs Reflection – negative scale factor fixed point
direction to scale
scale factor
longer (α>1) or smaller (0≤α<1)
Reflection – negative scale factor
effect of scale factor
reflection
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14. Transformations in Homogeneous Coordinates
Representations in homogeneous coordinates
Affine transformation – 44 matrix
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15. ? Translation Point p to p’ by displacing by a distance d
translation matrix
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16. Translation Point p to p’ by displacing by a distance d
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17. ? Translation Point p to p’ by displacing by a distance d
Inverse of a translation matrix ?
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18. Translation Point p to p’ by displacing by a distance d
Inverse of a translation matrix
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19. ? Scaling Scaling matrix with a fixed point of the origin
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20. Scaling Scaling matrix with a fixed point of the origin
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21. ? Scaling Scaling matrix with a fixed point of the origin
Inverse of a scaling matrix ?
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22. Scaling Scaling matrix with a fixed point of the origin
Inverse of a scaling matrix
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23. ? Rotation (1/2) Rotation with a fixed point at the origin
rotation matrix
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24. ? Rotation (1/2) Rotation with a fixed point at the origin
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25. ? Rotation (1/2) Rotation with a fixed point at the origin
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26. Rotation (1/2) Rotation with a fixed point at the origin
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27. Rotation (2/2)
Inverse of a rotation matrix ?
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28. Rotation (2/2) Inverse of a rotation matrix : orthogonal matrix
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29. shear the object in the x direction
One more affine transformation
shear the object in the x direction ?
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30. shear the object in the x direction
One more affine transformation
shear the object in the x direction
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31. ? Shear (2/2) Shear in the x direction shearing matrix
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32. Shear (2/2)
Shear in the x direction
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33. ? Shear (2/2) Shear in the x direction Inverse of a shearing matrix
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34. Shear (2/2) Shear in the x direction Inverse of a shearing matrix
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35. Concatenation of Transformations
Concatenating
affine transformations by multiplying together
sequences of the basic transformations
 define an arbitrary transformation directly
ex) three successive transformations p A B C q M p q
CBA
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36. Rotation about a Fixed Point (1/3)
Fixed point: pf
apply Rz() to rotation about a fixed point
rotation of a cube about its center
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37. Rotation about a Fixed Point (2/3)
sequence of transformations
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38. Rotation about a Fixed Point (3/3)
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39. General Rotation (1/2)
Three successive rotations about the three axes
rotation of a cube about the z axis
rotation of a cube about the y axis ?
rotation of a cube about the x axis
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40. General Rotation (2/2)
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41. Instance Transformation (1/2)
Instance of an object’s prototype
occurrence of that object in the scene
Instance transformation
applying an affine transformation to the prototype to obtain desired size, orientation, and location ?
instance transformation
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42. Instance Transformation (2/2)
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43. Rotation about an Arbitrary Axis (1/6)
Needs
fixed point: p0
rotation angle: θ
rotation axis: vector p2-p1
rotation of a cube about an arbitrary axis
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44. Rotation about an Arbitrary Axis (2/6)
First transformation is translation T(-p0) and the final one is T(p0)
Rotation problem!!!
we can get an arbitrary rotation
from three rotations about
individual axes
carry out two rotations to align
the axis of rotation with the z
axis
rotate by θ about the the z axis
movement of the fixed point to the origin
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45. Rotation about an Arbitrary Axis (3/6)
sequence of rotations
Determine x and y
direction angles and cosines
direction angles
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46. Rotation about an Arbitrary Axis (4/6)
Determine x and y (cont’)
projection line segment into plane y=0
look at the projection of line segment (before rotation) on the plane x=0
computation of the x rotation
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47. Rotation about an Arbitrary Axis (5/6)
Determine x and y (cont’)
projection line segment into z axis
rotation about y axis
caution!!! – clockwise angle
computation of the y rotation
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48. Rotation about an Arbitrary Axis (6/6)
Finally concatenate all the matrices
Ex) rotate an object by 45 degrees about the line passing through the origin and the point (1,2,3), fixed point is the origin
solution – textbook p.195~196
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49. OpenGL Transformation Matrices (1/2)
CTM (Current Transformation Matrix)
matrix that is applied to any vertex
part of the pipeline
: replacement
: initialization operation:
: postmultiplication
vertices
vertices
CTM
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50. OpenGL Transformation Matrices (2/2)
Matrix modes
model-view and projection matrices
Three functions: rotation, translation, scaling
vertices
vertices
model-view
projection
CTM
glRotatef(angle, vx, vy, vz);
glTranslatef(dx, dy, dz);
glScalef(sx, sy, sz);
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51. Example: Rotation about a Fixed Point in OpenGL
Needs
fixed point: (4, 5, 6)
rotation angle: 45 degrees
rotation axis: the line through the origin and the point (1,2,3)
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52. Example: Rotation about a Fixed Point in OpenGL
Needs
fixed point: (4, 5, 6)
rotation angle: 45 degrees
rotation axis: the line through the origin and the point (1,2,3)
glMatrixMode(GL_MODELVIWE);
glLoadIdentity( );
glTranslatef(4.0, 5.0, 6.0);
glRotatef(45.0, 1.0, 2.0, 3.0);
glTranslatef(-4.0, -5.0, -6.0);
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53. Rotating a Cube (1/2) glutDisplayFunc(display);
glutIdleFunc(spinCube);
glutMouseFunc(mouse);
void display(void) {
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glLoadIdentity();
glRotatef(theta[0], 1.0, 0.0, 0.0);
glRotatef(theta[1], 0.0, 1.0, 0.0);
glRotatef(theta[2], 0.0, 0.0, 1.0);
colorcube();
glutSwapBuffers(); }
void mouse(int btn, int state, int x, int y) {
if(btn==GLUT_LEFT_BUTTON && state==GLUT_DOWN) axis=0;
if(btn==GLUT_MIDDLE_BUTTON && state==GLUT_DOWN) axis=1;
if(btn==GLUT_RIGHT_BUTTON && state==GLUT_DOWN) axis=2; }
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54. Rotating a Cube (2/2) void spinCube(void) { theta[axis] += 2.0;
if( theta[axis] > ) theta[axis] -= 360.0;
glutPostRedisplay(); }
void mykey(char key, int mousex, int mousey) {
if( key == ‘q’ || key == ‘Q’ ) exit(); }
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55. Rotating a Cube (2/2) glPushMatrix( ), glPopMatrix( )
void spinCube(void) {
theta[axis] += 2.0;
if( theta[axis] > ) theta[axis] -= 360.0;
glutPostRedisplay(); }
void mykey(char key, int mousex, int mousey) {
if( key == ‘q’ || key == ‘Q’ ) exit(); }
glPushMatrix( ), glPopMatrix( )
perform a transformation and then return to the same state as before its execution
ex) instance transformation
glPushMatrix();
glTranslatef(.....);
glRotatef(.....);
glScalef(.....);
/* draw object here */
glPopMatrix();
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56. Virtual Trackball (1/3)
Use the mouse position to control rotation about two axes
Support continuous rotations of objects
trackball frame
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57. Virtual Trackball (2/3) Rotation with a virtual trackball
projection of the trackball position to the plane
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58. Virtual Trackball (3/3) Rotation with a virtual trackball (cont’)
determination of the orientation of a plane
rotation angle 
Quaternions
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59. Complex Numbers (1/3) Real part + imaginary part:
Addition and subtraction
Scalar multiplication
Multiplication y x z
imaginary
axis
real axis
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60. Complex Numbers (2/3) Imaginary unit: Complex conjugate Division
modulus or absolute value
Division
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61. Complex Numbers (3/3) Representation with polar coordinates
Euler’s formula
Complex multiplication and division
nth roots r θ
z=(x, y)
Imaginary
axis
real axis
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62. Quaternions (1/2) One real part + three imaginary part Properties:
Addition and scalar multiplication
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63. Quaternions (2/2) Ordered-pair notation Addition: Multiplication
scalar ‘s’ + vector “v = (a, b, c)”
Addition:
Multiplication
Magnitude
Inverse
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64. Quaternions and 3D Rotation
For a 3D point (α, β, γ)
a unit quaternion its conjugate 
For
Rq is a 3D rotation about (ux, uy, uz) by 2θ
Rotating (α, β, γ) by angle 2θ about the axis parallel to (ux, uy, uz)
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65. Rotations with Quaternions (1/2)
Rotation about any axis
set up a unit quaternion (u: unit vector)
represent any point position P in quaternion notation (p = (x, y, z))
carry out with the quaternion operation (q-1=(s, –v))
produce the new quaternion
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66. Rotations with Quaternions (2/2)
Obtain the rotation matrix by quaternion multiplication
Include the translations
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67. Example Rotation about z axis Set the unit quaternion:
Substitute a=b=0, c=sin(θ/2) into the matrix:
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