1. Advanced Graphics Algorithms Ying Zhu Georgia State University
Lecture 6
Transformations

2. Outline 3D transformations Concatenation of transformations
Transformations in Blender
Transformations in OpenGL
Anatomy of a simple OpenGL program

3. Geometric transformation
In 3D graphics, animation is achieved by transforming geometric objects in the virtual world
Transforming geometric objects means transforming points (vertices) and vectors
Transformation of points and vectors can be conveniently achieved by matrix multiplications
Introduced by Larry Roberts in 1966

4. Linear Algebra Review Matrix: An array of elements
Vectors: A n x 1 matrix is called a column vector
In computer graphics, vertex coordinates and vectors (directed line segments) and are represented by column vectors.

5. Linear Algebra Review
Matrix-column vector multiplication

6. Linear Algebra Review
Matrix-matrix multiplication V1 V2 V3

7. Linear Algebra Review Homogeneous representation
A vertex coordinate representation has 1 as the 4th component.
OpenGL uses homogeneous coordinates for all vertices.
(x, y, z) => (x, y, z, 1)
Why use homogeneous coordinates?
So that all transformations can be defined by matrix multiplication.

8. Affine Matrix An affine matrix is a 4x4 matrix with last low [0 0 0 1]
Why include the bottom row of (0,0,0,1)?
It’ll be useful in perspective transforms.
Affine matrix is used to define affine transformation.

9. Affine Transformation
Any transformation preserving
collinearity (i.e., all points lying on a line initially still lie on a line after transformation)
ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation).
Affine transformations preserve parallel lines.
We can transform vertices with affine transformation.
Affine transformations are what we use all the time in computer graphics.

10. Other Transformations
Isometry (preserves distance)
Similarity (preserves angles)
Collineation (lines remains lines)
Perspective projections
Non-linear (lines become curves)
Twists, bends, etc.
Can be used for deformation.

11. Elementary Affine Transformations
Four elementary affine transformations
Translation
Scaling
Rotation (about an axis)
Shear
Translation, scaling and rotation are the most commonly used transformations.

12. Translation
Translation P’ P d x y

13. Scaling
Scaling

14. Rotation Rotation about the origin
Counter-clock wise rotation by 90 degree.

15. 2D Transformation Summary
Translation
Scaling
Rotation

16. Composition of Affine Transforms
Most of the time we want to apply more than one transformation.
Example:
Translate by vector t
Rotate by r degree about the origin
This can be represented by multiple matrix multiplications:

17. Composition of Affine Transforms
Matrix multiplication is associative:
We can group the combined transformation as
The affine matrices are applied right side first.
We can also apply all the transformations to form a single matrix
This called composing or concatenation of transforms.
The result is another affine transformation.

18. Composition of Affine Transforms
It is important to always remember that matrix multiplication is not commutative
The order of transforms DOES matter
E.g. scaling then rotating is usually different than rotating then scaling
Keep this in mind when you write OpenGL program – you need to know in which order your OpenGL transform calls are executed

19. Why use matrix formalisms?
Can’t I just hardcode transforms rather than use the matrix multiplication?
Yes. But it’s harder to derive, harder to debug, and not any more efficient
All current graphics APIs use this matrix formalism
Must understand this to use OpenGL or other graphics libraries
The beauty of matrix formalism: the accumulated effect of many transformations can be stored in a single matrix

20. 3D Affine Transformations
3D affine transformations are an extension of the 2D transformations.
Use 4x4 matrices instead of 3x3 matrices in 2D transformations.
For example
3D scaling

21. 3D Translation 3D translation matrix
OpenGL function: glTranslatef(tx, ty, tz)

22. 3D Scaling
3D scaling matrix
OpenGL Function: glScalef(Sx, Sy, Sz)

23. 3D Rotation About the Origin
3D rotation is more complicated than 2D rotation.
Rotation about X axis
Rotation about Y axis
Rotation about Z axis

24. Arbitrary 3D Rotations
Normal practice is to decompose an arbitrary 3D rotation into rotations about x, y, z axes.
Create a single transformation matrix by composing.

25. Rotation in OpenGL
glRotatef(GLfloat angle, GLfloat x, GLfloat y, GLfloat z)
Equivalent to multiplying the following matrix

26. Rotation in OpenGL
Rotations are specified by a rotation axis and an angle
glRotatef(GLfloat angle, GLfloat x, GLfloat y, GLfloat z)
Angle is in degrees
x, y, and z are coordinates of a vector, the rotation axis
Normal practice is to decompose an arbitrary 3D rotation into rotations about x, y, z axes
Achieved by 3 glRotatef() calls

27. Rotation About a Fixed Point
The order is important:
Translate to the origin
Rotate about the origin
Translate back
Example:
To rotate by r about z around

28. Rotation About a Fixed Point
E.g. Rotate around point (9, -5, 3), 30 degree about z axis, then 45 degree about y axis …
glTranslatef(9.0, -5.0, 3.0);
glRotatef(45.0, 0.0, 1.0, 0.0);
glRotatef(30.0, 0.0, 0.0, 1.0);
glTranslatef(-9.0, 5.0, -3.0);
glBegin(GL_POINTS);
glVertex3f(…);
glEnd();

29. Multiple 3D Transformations
By combining a scaling, rotation, and a translation matrix we get a single transformation matrix.
Notice that in the single matrix, translation is separated from scaling and rotation part.

30. Transformations in OpenGL
Internally each vertex in OpenGL is represented as homogeneous points
i.e. a column vector with 1 as the last item.
However you don’t need to specify the 1 in your program
Each vertex is transformed by the modelview matrix.
There is always a current modelview matrix that is applied to all vertices.
First tell OpenGL you’ll modify the modelview matrix by calling glMatrixMode(GL_MODELVIEW);
You need to initialize the modelview matrix to the identity matrix through glLoadIdentity();

31. Model-view Matrix

32. Manipulating the Current Matrix
Two ways to modify the modelview matrix:
Use OpenGL transformation calls (internally generate a matrix)
Explicitly load pre-defined affine matrix as the current modelview matrix.
We have to first tell OpenGL we’ll modify the modelview matrix by calling
glMatrixMode(GL_MODELVIEW);

33. OpenGL Transformation Calls
Translation:
glTranslatef(dx, dy, dz);
Scaling:
glScalef(sx, sy, sz);
Rotation:
glRotatef(angle, vx, vy, vz);
Rotate object counter-clock wise around vector (vx, vy, vz)
Angle is in degrees not radians.
There is a matrix multiplication behind every transformation function call
Current_MV_matrix  Current_MV_matrix * matrix

34. Example Pay attention to the patterns:…
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(10.0, 20.0, 0.0);
glRotatef(30.0, 0.0, 1.0, 0.0);
glBegin(GL_LINES);
glVertex3d(0, 0, 0);
glVertex3d(100, 120, 0);
glEnd();
Pay attention to the patterns:
Specify transformations first, and then draw 3D geometries.
The order of transformations: rotate first, translate later. Transformations are applied in the reverse order – last transformation is applied first!
This is what makes OpenGL difficult to learn.

35. Transformation in OpenGL
It is important to understand the relationship among
Current modelview matrix,
OpenGL transformation calls
Vertices of 3D Object
So you can understand why OpenGL code has to be arranged in certain way

36. What happens inside OpenGL during transformation?
Final modelview matrix
Matrices created by OpenGL transformation calls
Initial modelview matrix (initialized as an identity matrix)
glMatrixMode(GL_MODELVIEW)
glLoadIdentity()
Vertex coordinates before transformation
glVertex3f()
Vertex coordinates after transformation

37. OpenGL matrix stack
OpenGL maintains a matrix stack to temporarily store a “snapshot” of the current modelview matrix.
We often need to back up the current modelview matrix, modify it, and then load the previous modelview matrix.
glPushMatrix(); // Push the current matrix onto stack
glPopMatrix(); // Pops the stack and load the top matrix // to be the current modelview matrix.
This makes it possible to do hierarchical transformation
glPushMatrix() and glPopMatrix() can be used to control the scope of transformation calls

38. Why use glPushMatrix()/glPopMatrix()?…
glRotatef(theta[0], 0.0, 1.0, 0.0); // apply to all subsequent objects
Draw_torso();
glPushMatrix();
glTranslatef(0.0, HEADX, 0.0); // only apply to head
glRotatef(theta[1], 1.0, 0.0, 0.0); // only apply to head
Draw_head();
glPopMatrix();
glRotatef(theta[3], 1.0, 0.0, 0.0); // apply to left_upper_arm and left_lower_arm
Draw_left_upper_arm();
glTranslatef(0.0, LLAY, 0.0); // apply only to left_lower_arm
glRotatef(theta[4], 1.0, 0.0, 0.0); // apply only to left_lower_arm
Draw_left_lower_arm();
Draw_somthing_else(); // which transform(s) call is applied to this one?

39. Anatomy of an OpenGL program/*
* planet.c
* This program shows how to composite modeling transformations
* to draw translated and rotated models.
* Interaction: pressing the d and y keys (day and year)
* alters the rotation of the planet around the sun. */
#include <GL/glut.h>
#include <stdlib.h>
static int year = 0, day = 0;
void init(void) {
glClearColor (0.0, 0.0, 0.0, 0.0); // initialize frame buffer to black color (background)
glShadeModel (GL_FLAT); // choose shading model }

40. Anatomy of an OpenGL program
void display(void) {
glClear (GL_COLOR_BUFFER_BIT);
glColor3f (1.0, 1.0, 1.0);
glPushMatrix(); // current modelview matrix is the view matrix
glutWireSphere(1.0, 20, 16); // draw sun, internally call glVertex3f()
glRotatef ((GLfloat) year, 0.0, 1.0, 0.0); // Multiply modelview matrix with a // rotation matrix
glTranslatef (2.0, 0.0, 0.0); // Multiply current modelview matrix with a
// translation matrix
glRotatef ((GLfloat) day, 0.0, 1.0, 0.0); // Multiply current modelview matrix with a
// rotation matrix
glutWireSphere(0.2, 10, 8); // draw smaller planet, internally call glVertex3f()
glPopMatrix(); // current modelview matrix is view matrix again
glutSwapBuffers(); }

41. Anatomy of an OpenGL program
void reshape (int w, int h) {
glViewport (0, 0, (GLsizei) w, (GLsizei) h); // adjust image size based on // window size
glMatrixMode (GL_PROJECTION);
glLoadIdentity ();
// initialize projection matrix
gluPerspective(60.0, (GLfloat) w/(GLfloat) h, 1.0, 20.0);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(); // initialize modelview matrix to identity matrix
// place camera, internally generates a view matrix and
// then multiply the current modelview matrix with the view matrix
gluLookAt (0.0, 0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0); }

42. Anatomy of an OpenGL program
void keyboard (unsigned char key, int x, int y) {
switch (key) {
case 'd':
day = (day + 10) % 360;
glutPostRedisplay(); // force a call to display()
break;
case 'D':
day = (day - 10) % 360;
glutPostRedisplay();
case 'y':
year = (year + 5) % 360;
case 'Y':
year = (year - 5) % 360;
case 27:
exit(0);
default: }

43. Anatomy of an OpenGL program
int main(int argc, char** argv) {
glutInit(&argc, argv);
glutInitDisplayMode (GLUT_DOUBLE | GLUT_RGB);
glutInitWindowSize (500, 500);
glutInitWindowPosition (100, 100);
glutCreateWindow (argv[0]);
init (); // clear frame buffer and select shading model
glutDisplayFunc(display); // register event handlers
glutReshapeFunc(reshape);
glutKeyboardFunc(keyboard);
glutMainLoop(); // Start the event loop
return 0; }

44. Summary Affine transformations are used for transforming 3D objects.
Three elementary affine transformations:
Translation
Scaling
Rotation
3D rotation is the most complicated among the three.
If we use homogeneous coordinates, then all affine transformations can be represented by affine matrix multiplication.
Notice the pattern of transformation in OpenGL programming.

45. OpenGL Tutorial Programs
Download OpenGL transformation tutorial program from

46. Readings OpenGL Programming Guide: Chapter 3 Sample programs
Cube.c, planet.c, robot.c