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Composing Transformations

(pre-compute). M. 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:

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Composing Transformations

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  1. (pre-compute) M 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

  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. y x z How OpenGL does it? • Rotation: • glRotatef(d)(angle, vx, vy, vz) -> glRotatef(d)(angle, 0,0,1) for 2D 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 useris responsible to reset it if necessary glLoadIdentity() -> M = 1 0 0 0 1 0 0 0 1

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

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

  9. Wrong!!! Something noteworthy • Very very noteworthy … • 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

  10. glTranslated(5,0,0); glRotate(60,0,0,1); glBegin() … glRotated(60,0,0,1); glTranslated(5,0,0); glBegin() Example Revisit • We want rotation and then translation • Generate wrong results if you do: 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 Translate (3,3)?

  14. Translate Coordinate Frame (2) Translate (3,3)?

  15. 30 degree Rotate Coordinate Frame Rotate 30 degree?

  16. Scale Coordinate Frame Scale (0.5,0.5)?

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

  18. o 60 (5,5) Another example 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) ??? Answer: Translate(5,5) and then Rotate (60)

  19. o o 60 60 Another example (cont’d) If you Rotate(60) and then Translate(5,5) … C2 C1 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!!!

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