1. Geometric Transformations
Basic 2D geometric transformation
Homogeneous coordinates
2D transformations
Translation
Scaling
Rotation
Composite transformations
Other transformation: Shearing, Reflection
3D transformations
Other 3D transformation: Shearing, Reflection
OpenGL Matrix stack
Lecture 7 Transformation

2. 1. Basic 2D Transformation
2D Translation A translation moves a vertex to a new position by adding a displacement vector to it.
Eg, The following translates the vertex to by adding
(3,13)
(7,7)
Lecture 7 Transformation

3. 2D Rotation r   (x’,y’) (x,y) x=r cos y=r sin x’ = r cos(+)
= r cos cos  r sin sin
= x cos - y sin
y’ = r sin(+)
= r sin cos + r cos sin
= x sin + y cos
y=r sin
Lecture 7 Transformation

4. 2D Scaling x’ = x sx y’ = y sy (x, y) (x’, y’)
Lecture 7 Transformation

5. 2. Linear Transformations on vertices
A linear transformation moves a vertex to a new position by multiplying it with a non-singular matrix.
Eg, The following transforms the vertex (2, 3) to (3, 13)
(3,13)
(2,3)
Lecture 7 Transformation

6. Affine Transformations on vertices
An affine transformation is a linear transformation followed by a translation. Its 2D general form is
Affine transformations preserve lines.
Many geometric movements of objects, eg, translations, rotations, and scalings are affine transformations.
Lecture 7 Transformation

7. 3. Homogeneous coordinates
A point in homogeneous coordinates (x, y, w), w ≠ 0, corresponds to the 2-D vertex (x/w, y/w) in Cartesian coordinates
Conceive that the Cartesian coordinates axes lies on the plane of w = 1. The intersection of the plane and the line connecting the origin and (x, y, w) gives the corresponding Cartesian coordinates w
(x, y, w)
w = 1
(x/w, y/w, 1) x y y x
w = 0
Lecture 7 Transformation

8. E.g. both the points (6, 9, 3) and (4, 6, 2) in the homogeneous coordinates corresponds to (2, 3) in the Cartesian coordinates.
Conversely, the point (2, 1) of the Cartesian corresponds to (2, 1, 1), (4, 2, 2) or (6, 3, 3) of the homogeneous w
6, 3, 3)
(4, 2, 2)
w = 1
(2, 1, 1) x y y x
w = 0
Lecture 7 Transformation

9. Homogeneous coordinates of vectors
A 2-D vector (x, y) of Cartesian corresponds to (x, y, 0) in homogeneous coordinates and vice versa.
Note that vectors and vertices have different representations in homogeneous coordinates.
On the other hand, the transformation formulas for both vectors and vertices in homogeneous coordinates are identical. Such generalization enhances simplicity, and thus the reliability, of a graphics system
Lecture 7 Transformation

10. 4. 2D Transformation in Homogeneous form
2D Translation in Homogeneous form
(1,1)
(5,1)
(1,3)
(x,y)
(3,2)
(7,2)
(3,4)
(x’, y’)
Lecture 7 Transformation

11. 2D Rotation in Homogeneous form
Lecture 7 Transformation

12. Eg, to rotate 45o (/4 radian) sin /4 = cos /4 = 0.7071
(1,1)
(5,1)
(1,3)
(x,y)
(0,1.4)
(2.8, 4.2)
(-1.4, 2.8)
(x’, y’)
Lecture 7 Transformation

13. 2D Translation in Homogeneous
(1,1)
(5,1)
(1,3)
(x,y)
(2,1/2)
(10,1/2)
(2,3/2)
(x’, y’)
Lecture 7 Transformation

14. 5. Composite a sequence of transformations
Translate the right-angle vertex
to the origin (Tx = -1, Ty = -1)
Rotate 45o (/4 radian)
sin /4 = cos /4 =
(1,1)
(5,1)
(1,3)
(0,0)
(4,0)
(0,2)
(2.8,2.8)
(-1.4,1.4)
Lecture 7 Transformation

15. Lecture 7 Transformation

16. 6. Composite Transformation Implementation
All practical transformations on vertices and vectors in homogeneous coordinates can be expressed as pure matrix multiplications. Such uniformity greatly enhance the run-time efficiency of a graphics system.
Suppose that a set of 1 million vertices needs to go through three transformations representing by the matrices M1, M2, and M3. A straightforward formula of the overall transformations for a vertex v is v´ = M3*M2*M1*v.
As matrix multiplication is associative, we can compute M = M3*M2*M1 in advance, then transform all 1 million vertices using the formula v´ = M*v.
Lecture 7 Transformation

17. 7. Other 2D Transformation
Shearing
Uneven scaling and shearing change the shape of an object
An example of 2D shearing in x direction (To alter x-coordinates by an amount proportional to the y value.) y x
(0,0)
(0,1)
(1,0)
(1,1)
(2,1)
(3,1)
x’ = x + 2y
y’ = y
Lecture 7 Transformation

18. x’ = x y’ = 2x + y y x y x (0,0) (0,1) (1,2) (1,3) (0,0) (0,1) (1,0)
(1,1)
x’ = x
y’ = 2x + y
Lecture 7 Transformation

19. (x,y) (x,-y) Reflection about the x axis Reflection about the y axis ?
Reflection w.p.t the origin ?
Reflection w.p.t the x = y
Lecture 7 Transformation

20. y =mx+b Reflection along a line
Translate (0, -b) so that the line
passes through the origin
Rotate the line onto the x axis by -o
Reflect about the x axis
Backward rotate
backward translate
(Be reminded that these operations must be specified in reverse order.) b 
Lecture 7 Transformation