1. Transformations CS4395: Computer Graphics 1 Mohan Sridharan Based on slides created by Edward Angel

2. Objectives Introduce standard transformations: – Rotation. – Translation. – Scaling. – Shear. Derive homogeneous coordinate transformation matrices. Build arbitrary transformation matrices from simple transformations. CS4395: Computer Graphics 2

3. General Transformations A transformation maps points to other points and/or vectors to other vectors. CS4395: Computer Graphics 3 Q=T(P) v=T(u)

4. Affine Transformations Preserves lines. Characteristic of many important transformations: – Rigid body transformations: rotation, translation. – Scaling, shear. In CG: Only need to transform end-points of line segments. Let implementation draw line segment between the transformed endpoints. CS4395: Computer Graphics 4

5. Pipeline Implementation v CS4395: Computer Graphics 5 transformationrasterizer u u v T T(u) T(v) T(u) T(v) vertices pixels frame buffer (from application program)

6. Notation Will be working with coordinate-free representations of transformations and representations within a particular frame. P,Q, R: points in an affine space. u, v, w : vectors in an affine space; , ,  : scalars. p, q, r : representations of points: -array of 4 scalars in homogeneous coordinates. u, v, w : representations of points: -array of 4 scalars in homogeneous coordinates. CS4395: Computer Graphics 6

7. Translation Move (translate, displace) a point to a new location: Displacement determined by a vector d: – Three degrees of freedom. – P’=P+d. CS4395: Computer Graphics 7 P P’ d

8. How many ways? Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way CS4395: Computer Graphics 8 objecttranslation: every point displaced by same vector

9. Translation Using Representations Using the homogeneous coordinate representation in some frame: p=[ x y z 1] T p’=[x’ y’ z’ 1] T d=[dx dy dz 0] T Hence p’ = p + d or: x’=x+d x y’=y+d y z’=z+d z CS4395: Computer Graphics 9 note that this expression is in four dimensions and expresses point = vector + point

10. Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates: p ’= Tp where T = T (d x, d y, d z ) = This form is better for implementation: – All affine transformations can be expressed this way. – Multiple transformations can be concatenated together. CS4395: Computer Graphics 10

11. Rotation (2D) Consider rotation about the origin by  degrees: – radius stays the same, angle increases by  CS4395: Computer Graphics 11 x’=x cos  –y sin  y’ = x sin  + y cos  x = r cos  y = r sin  x = r cos (  y = r sin ( 

12. Rotation about the z axis Rotation about z axis in three dimensions leaves all points with the same z. – Equivalent to rotation in 2D in planes of constant z: – In homogeneous coordinates: p ’= R z (  )p CS4395: Computer Graphics 12 x’=x cos  –y sin  y’ = x sin  + y cos  z’ =z

13. Rotation Matrix (about z-axis) R = R z (  ) = CS4395: Computer Graphics 13

14. Rotation about x and y axes Same argument as for rotation about z axis: – For rotation about x axis, x is unchanged. – For rotation about y axis, y is unchanged. CS4395: Computer Graphics 14 R = R x (  ) = R = R y (  ) =

15. Scaling S = S(s x, s y, s z ) = CS4395: Computer Graphics 15 x’=s x x y’=s y x z’=s z x p’=Sp Expand or contract along each axis (fixed point of origin)

16. Reflection Corresponds to negative scale factors! CS4395: Computer Graphics 16 original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

17. Inverses Though general formulae can be used, simple geometric observations help: – Translation: T -1 (d x, d y, d z ) = T (-d x, -d y, -d z ) – Rotation: R -1 (  ) = R(-  ) Holds for any rotation matrix. Note that since cos(-  ) = cos(  ) and sin(-  )=-sin(  ) R -1 (  ) = R T (  ) – Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z ) CS4395: Computer Graphics 17

18. Concatenation Form arbitrary affine transformation matrices by multiplying rotation, translation, and scaling matrices. Because the same transformation is applied to many vertices, the cost of forming matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p. The challenge: how to form a desired transformation from the specifications in the application? CS4395: Computer Graphics 18

19. Order of Transformations Matrix on the right is applied first. Mathematically, the following are equivalent: p’ = ABCp = A(B(Cp)) Many references use column matrices to represent points: p ’T = p T C T B T A T CS4395: Computer Graphics 19

20. General Rotation About the Origin  CS4395: Computer Graphics 20 x z y v A rotation by  about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes! R(  ) = R z (  z ) R y (  y ) R x (  x )  x  y  z are called the Euler angles Note that rotations do not commute! Can use rotations in another order but with different angles.

21. Rotation About a Point other than the Origin Move fixed point to origin. Rotate. Move fixed point back. M = T(p f ) R(  ) T(-p f ) CS4395: Computer Graphics 21

22. Instancing In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size. We apply an instance transformation to its vertices to scale, orient, locate. CS4395: Computer Graphics 22

23. Shear Helpful to add one more basic transformation. Equivalent to pulling faces in opposite directions. CS4395: Computer Graphics 23

24. Shear Matrix Consider simple shear along x axis: CS4395: Computer Graphics 24 x’ = x + y cot  y’ = y z’ = z H(  ) =