1. Geometric Transformations

2. Geometric Transformations
Translate
X’ = X + dx
Y’ = Y + dy
X’ = X + (-6)
Y’ = Y + (-4) dy dx

3. Geometric Transformations
Scale
X’ = X * Sx
Y’ = Y * Sy
X’ = X * 2
Y’ = Y * 0.5
Only origin is stable

4. Geometric Transformations
Scale
X’ = X * Sx
Y’ = Y * Sy
X’ = X * -1
Y’ = Y * 1
Only origin is stable

5. Geometric Transformations
Rotate
X’ = cos(a)X – sin(a)Y
Y’ = sin(a)X + cos(a)Y
X’ = cos(45)X – sin(45) Y
Y’ = sin(45)X + cos(45)Y
Only origin is stable

6. General Form Translate Scale Rotate X’ = 0 * X + 0 * Y + dx
Y’ = 0 * X + 0 * Y + dy
Scale
X’ = Sx * X + 0 * Y + 0
Y’ = 0 * X + Sy * Y + 0
Rotate
X’ = cos(a)*X – sin(a)*Y + 0
Y’ = sin(a)*X + cos(a)*Y + 0

7. Homogenous Coordinates
Translate
X’ = 1 * X + 0 * Y + dx = [1, 0, dx] * [ X, Y, 1]
Y’ = 0 * X + 1 * Y + dy = [0, 1, dy] * [ X, Y, 1]
Scale
X’ = Sx * X + 0 * Y + 0 = [Sx, 0, 0] * [ X, Y, 1]
Y’ = 0 * X + Sy * Y + 0 = [0, Sy, 0] * [ X, Y, 1]
Rotate
X’ = cos(a)*X – sin(a)*Y + 0
= [cos(a), -sin(a), 0] * [ X, Y, 1]
Y’ = sin(a)*X + cos(a)*Y + 0
= [sin(a), cos(a), 0] * [ X, Y, 1]

8. Matrix form [ X Y 1] * a d 0 = [X’ Y’ 1] b e 0 c f 1 a b c * X = X’
d e f Y Y’

9. Matrix form 1 0 dx * X = X’ 0 1 dy Y Y’ 0 0 1 1 1 T(dx,dy) S(Sx, Sy)
T(dx,dy)
S(Sx, Sy)
R(a)
R(sin(a),cos(a))
Sx 0 0 * X = X’
0 Sy 0 Y Y’
cos(a) -sin(a) 0 * X = X’
sin(a) cos(a) 0 Y Y’

10. T(2, 2) S(3/4, 1/3) T(-2,-2) 3/
0 1/3 0 X Y 1 X’ Y’ 1 =

11. T(2, 2) S(3/4, 1/3) T(-2,-2) 3/
0 1/3 0 X Y 1 X’ Y’ 1 =

12. R(45) S(2,1)
S(2,1) R(45)