Chapter 6
Geometrical Transformations

6.1 Basic Transformations

• Translation
  x‘	= x + Tx
  y‘	= y + Ty

  
• Scaling about the origin
  x‘	= Sx . x
  y‘	= Sy . y

  
• Rotation about the origin
  x‘	= R cos( + )
  y‘	= R sin( + )

  x	= R cos
  y	= R sin

  ____________________
  cos( + )	= cos cos - sin sin
  sin( + )	= sin cos + cos sin

  ______________________
  x‘	= xcos - y sin
  y‘	= y cos + xsin

6.2 Composite Transformations

Transformations that can be decomposed into basic transformations.

• Scaling about arbitrary point.

  

• Rotation about arbitrary point

  

6.3 Matrices

Tools for representing transformations.

• MATRIX -- A rectangular collection of elements.

  

• SUM -- A + B = (a_ij + b_ij)

  + =

• PRODUCT -- A ^. B = ( _k=1ⁿa_ikb_kj)

  ^. =

• ASSOCIATIVE addition and product (order of execution is insignificsnt).
  A + (B + C)	= (A + B) + C
  A . (B . C)	= (A . B) . C

• Product is normally NONCOMMUTATIVE (i.e., order of operands is important)

6.4 Matrix Representations for Transformations

• Translation by (T_x , T_y)
  x‘	= x + Tx
  y‘	= y + Ty

  +
• Scaling by (S_x , S_y ) about the origin
  x‘	= Sx . x
  y‘	= Sy . y

  ^.
• Rotation about the origin
  x‘	= xcos - y sin
  y‘	= xsin + y cos

  ^.
• Transformations are applied many times for an object (e.g., on 7 points in the following object).

  

• Compute composite transformations once ahead for all points before applying them on the points, instead of applying them in succession on the points.
  • Composite of translations:

    T_k + (... + (T₂ + (T₁ + P))...) = (T_k + ... + T₂ + T₁) + P

    + = + = +
  • Composite of scalings/rotations:

    X_k (...(X₂ ^. (X₁ ^. P))...) = (X_k ^. ... ^. X₂ ^. X₁) ^. P

    ^. = ^. = +

• Composite translation + scaling/rotation?

  = (?) ^.

6.5 Homogeneous Coordinates

• Homogeneous coordinates.

  (x,y,w) instead of (x, y)

• Points (x, y) in the plane z = w of a 3D space.

  

• Normalized homogeneous coordinates.

  (x,y,w) with w = 1

We’ll use only normalized homogeneous coordinates.

6.6 Homogeneous Transformations

• Translation by (T_x , T_y)

  ^. =

• Scaling by (S_x , S_y ) about the origin

  ^. =

• Rotation about the origin

  ^. =

• Scaling about point (t_x,t_y).

  T(t_x , t_y ) ^. S(s_x , s_y ) ^. T(-t_x,-t_y) =

  ^..=

• Rotation about point (t_x,t_y).

  T(t_x , t_y ) ^. R() ^. T(-t_x,-t_y) =

  ^..=