3. Translation
•Translation transformation
•Translation vector or shift vector T = (tx, ty)
•Rigid-body transformation
•Moves objects without deformation
y y
P’
T T
p
x x
4. Rotation
Rotation transformation (anticlockwise)
x’=rcos(θ+Φ)= rcosθcosΦ - rsin θsinΦ
y’=rsin(θ+Φ)= rcosθsinΦ + rsin θcosΦ y
x=rcosΦ y=rsin Φ P’ (x’,y’)
x’=x cosθ - ysinθ
y’=xsinθ+ycosθ θ P(x,y)
r
Φ x
P’= R· P
5. Rotation
General Pivot point rotation
x’=xr+(x- xr)cosθ - (y- yr)sinθ y
P’ (x’,y’)
y’=yr+(x- xr)sinθ+(y- yr)cosθ
θ P(x,y)
r
Φ
(xr,yr)
x
17. Concatenation Properties
Matrix multiplication is associative.
A·B ·C = (A·B )·C = A·(B ·C)
Transformation products may not be commutative
Be careful about the order in which the composite matrix is
evaluated.
Except for some special cases:
Two successive rotations
Two successive translations
Two successive scalings
rotation and uniform scaling
19. Reflection
A transformation produces a mirror image of an object.
Axis of reflection
A line in the xy plane
A line perpendicular to the xy plane
The mirror image is obtained by rotating the object 180 0 about the
reflection axis.
Rotation path
Axis in xy plane: in a plane perpendicular to the xy plane.
Axis perpendicular to xy plane: in the xy plane.