1. Composing Transformations
Composing Transformations – the process of applying several transformations in succession to form one overall transformation
If we transform a point P using matrix M1 first, and then transform using M2, and then M3, we have:
(M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P M
(pre-compute)

2. Composing Transformation
Matrix multiplication is associative
M3 x M2 x M1 = (M3 x M2) x M1 = M3 x (M2 x M1)
Transformation products may not be commutative A x B != B x A
Some cases where A x B = B x A
A B
translation translation
scaling scaling
rotation rotation
uniform scaling rotation
(sx = sy)

3. Transformation order matters!
Example: rotation and translation are not commutative
Translate (5,0) and then Rotate 60 degree OR
Rotate 60 degree and then translate (5,0)??
Rotate and then translate !!

4. How OpenGL does it?
OpenGL’s transformation functions are meant to be used in 3D
No problem for 2D though – just ignore the z dimension
Translation:
glTranslatef(d)(tx, ty, tz) -> glTranslatef(d)(tx,ty,0) for 2D

5. How OpenGL does it? Rotation:
glRotatef(d)(angle, vx, vy, vz) -> glRotatef(d)(angle, 0,0,1) for 2D x y z y
(vx, vy, vz) – rotation axis x
You can imagine z is pointing out
of the slide

6. OpenGL Transformation Composition
A global modeling transformation matrix
(GL_MODELVIEW, called it M here)
glMatrixMode(GL_MODELVIEW)
The user is responsible to reset it if necessary
glLoadIdentity()
-> M =
0 1 0
0 0 1

7. OpenGL Transformation Composition
Matrices for performing user-specified transformations are multiplied to the current matrix
For example,
glTranslated(1,1 0); M = M x
All the vertices defined within glBegin() / glEnd() will first go through the transformation (modeling transformation)
P’ = M x P

8. Transformation Pipeline
Object
Local Coordinates
Object
World Coordinates
Modeling
transformation …

9. Something noteworthy Very very noteworthy … Wrong!!!
OpenGL post-multiplies each new transformation matrix
M = M x Mnew
Example: perform translation, then rotation
0) M = Identity
1) translation T(tx,ty,0) -> M = M x T(tx,ty,0)
2) rotation R(q) -> M = M x R(q)
3) Now, transform a point P -> P’ = M x P
= T(tx, ty, 0) x R(q) x P
Wrong!!!

10. Example Revisit We want rotation and then translation
Generate wrong results if you do:
glRotated(60,0,0,1);
glTranslated(5,0,0);
glBegin()
glTranslated(5,0,0);
glRotate(60,0,0,1);
glBegin() …
You need to specify the transformation
in the opposite order!!

11. How Strange … OpenGL has its reason …
It wants you to think of transformation in a different way
Instead of thinking of transforming the object in a fixed global coordinate system, you should think of transforming an object as moving (transforming) its local coordinate system

12. OpenGL Transformation
When using OpenGL, we need to think of object transformations as moving (transforming) its local coordinate frame
All the transformations are performed relative to the current coordinate frame origin and axes

13. Translate Coordinate Frame

14. Translate Coordinate Frame (2)

15. Rotate Coordinate Frame
Rotate 30 degree?
30 degree

16. Scale Coordinate Frame

17. Compose Transformations
Answer:
Translate(7,9)
Rotate 45
Scale (2,2) o 45
(7,9)

18. Another example Answer: Translate(5,5) and then Rotate (60)
How do you transform from C1 to C2? C1 C2
Translate (5,5) and then Rotate (60) OR
Rotate (60) and then Translate (5,5) ???
(5,5) 60 o
Answer: Translate(5,5) and then
Rotate (60)

19. Another example (cont’d)
If you Rotate(60) and then Translate(5,5) … 60 o C1 C2 60 o 5 5
You will be translated (5,5)
relative to C2!!

20. Transform Objects
What does coordinate frame transformations have to do with object transformations?
You can view transformations as tying the object to a local coordinate frame and moving that coordinate frame

21. Put it all together When you use OpenGL …
Think of transformation as moving coordinate frames
Call OpenGL transformation functions in that order
OpenGL will actually perform the transformations in the reverse order
Everything will be just right!!!