1. Transformations of Objects – 3D
CS430 Computer Graphics
Transformations of Objects – 3D
Chi-Cheng Lin, Winona State University

2. Topics 3D Affine Transformations
Composition of 3D Affine Transformations
Properties of Affine Transformations
Changing Coordinate Systems

3. 3D Affine Transformations
Translation
T(dx, dy, dz) =

4. 3D Affine Transformations
Scaling
S(Sx, Sy, Sz) =

5. 3D Affine Transformations
Rotations: Three elementary rotations
Rotation about the x-axis (x-roll)
Rotation about the y-axis (y-roll)
Rotation about the z-axis (z-roll)
Positive values of rotation angel cause a counterclockwise (CCW) rotation about an axis as one looks inward from a point on the positive axis toward the origin

7. 3D Rotations
Let c = cos() and s = sin()
 x-roll
 z-roll
 y-roll

8. Composition of 3D Affine Transformations
Similar to 2D
If a sequence of transformations, represented by M1, M2, …, Mn, is applied to a 3D point P, then P is transformed to MP, where M = Mn    M2  M1
Important distinction between 2D and 3D rotations
3D rotation matrices do not commute
3D rotation can be about different axis

9. Combining 3D Rotations Euler’s Theorem Calculation is complex
Any rotation (or sequence of rotations) about a point is equivalent to a single rotation about some axis through that point
Calculation is complex
Good news: OpenGL can do it for you
glRotated(angle, ux, uy, uz),
rotates angle degrees about an axis with unit vector (ux, uy, uz)

11. Properties of Affine Transformations
Affine transformation preserve affine combinations of points
Affine transformation preserve lines and planes
Parallelism of lines and planes is preserved
Relative ratios are preserved
Every affine transformation is composed of fundamental transformations

12. Properties of Affine Transformations
If M is the 2D transformation matrix, then
Rotation, translation, and shearing do not change the area (why?)
For scaling, new area = (old area)|SxSy|
If M is the 3D transformation matrix, then

13. Properties of Affine Transformations
The columns of the matrix M reveal the transformed coordinate frame
In 2D,
The axes of the new coordinate frame are m1 and m2, and the origin is m3
The axes of new coordinates frame are not necessarily perpendicular nor must they be of unit length

14. Changing Coordinate Systems
Suppose coordinate system #2 is formed from coordinate system #1 by affine transformation M , then for a point P in system #2, its coordinates are MP.
System #2 P b
System #1 d c j’ i j i’ a

15. Changing Coordinate Systems
Successive changes in coordinate system
System #3 P b f
System #2 e
System #1 d c
System #1
System #2
System #3 M1 a M2

17. Changing Coordinate Systems
Transforming points
If a sequence of transformations, represented by M1, M2, …, Mn, is applied to a point P, then P is transformed to MP, where
M = Mn    M2  M1
Transforming coordinate system
If a sequence of transformations represented by M1, M1, …, Mn, is applied to the coordinate system, then a point P expressed in the transformed system has coordinates MP in the original system, where
M = M1  M2    Mn