Transformations

1
Transformations

• Consider transformation of a Unit Square
• Ability to transform points and lines
• Origin at A0,0, B, C, D figure 2.6a
• Apply the 2x2 matrix
• Origin is unaffected by transformation
• Coordinates of B equal to first row and D
  equals that of the second row of transform.

2
Unit Square Tranformation

• B and D determine the 2D transform
• Transformed figure is a parallelogram ( recall
  that parallel lines transform as parallel lines)
• Combination of shearing ( b and c terms) and
  scaling (a and d terms)
• Area property is of special interest why?

3
Area properties

• Ap area of big rectangle area of triangles(2)
  area of polygons(2)
• Area as function of determinant of the
  transformation matrix
• Ap As(ad bc) As area of initial square
• Generalisation as At Ai(ad bc)
• Area of arbitrary shapes in 2D

4
Arbitrary rotation

• Pure rotation of a unit square
• Consider rotation about the origin (fig 2.7)
• Positive rotation in the CCW direction
• Point B to B with x (1)cos _at_ and y
  (1) sin _at_
• Point D to D with x -(1) sin _at_ and y
  (1) cos _at_
• Point B and D define the arbitrary matrix

5
2x2 Rotation matrix

• Roation matrix is defined as
• cos _at_ sin _at_
• -sin _at_ cos _at_
• For 90 degree CCW rotation about the origin this
  reduces to 0 1
• -1 0
• Scaling and Shearing with no dimension change
  hence a pure rotation matrix ( _at_ )

6
Translations

• We have not introduced any translations
• Forrest (2-2) introduced third component to
  point vectors to overcome this
• x y 1 and x y 1
• The 3x3 transformation is introduced as follows
  1 0 0
• 0 1 0
• m n 1

7
Homogeneous Coordinates

• x y 1 is now the transformed vector
• Addition of third element to position vector
• Addition of third column/row to transformation
  matrix
• Look at this as an additional coordinate of the
  position vector
• x y 1 becomes X Y H

8
Homogeneous Coordinates

• Transformation to the plane H 1
• If the transformation is 1 0 p
• 0 1
  q
• m n
  s
• X Y H where H is not equal to 1
• Coming back to 2D, we restrict H 1
• x y X/H Y/H 1

9
Homogeneous Coordinates

• Representation of a n-component vector by an n1
  component vector is called homogeneous coordinate
  representation
• Do transformation in n1 space and project back
  to the n-dimensional space of interest (H 1 in
  our case)
• Physical coordinates are x hx/h and yhy/y

10
Uniqueness

• No unique homogeneous representation of a point
  in 2-space
• For example 12 8 4, 6 4 2 and 3
  2 1 all represent the physical coordinates of
  3 2
• In homogeneous coordinates X Y H x y 1
  
• a b 0
• c d 0
• 0 0 1

11
Normalization

• All transforms of x and y fall within the plane
  of H1 after normalization
• Translation and projection could now be
  introduced
• Scaling and shearing using a, b, c, d
• What is projection? x y (pxqy1) . See
  eq.2-34. Page 39)

12
Projection

• H px qy 1 defines the plane containing
  homogeneous coordinates
• Figure 2-9. AB to CD and back to C D
• Effect of normalizing is to project CD to C D
  letting H 1 using origin as center of
  projection
• Four parts of a 3x3 transformation matrix

13
3x3 TRANSFORM

• A,B,C,D produce scaling, shear and rotation
  (see page 40)
• M and N produce translation
• P and Q produce projection
• S produces overall scaling s lt 1 enlargement
  scaling or s gt 1 reduction scaling

14
Points at infinity

• Homogeneous coordinates will allow mapping one
  set of points to map from one coordinate system
  to another.
• Infinite to finite range
• x y 0 in 2-dimensions represents points at
  infinity
• Generalized transformation is of more practial
  use in CAE

15
ARBITRARY AXIS ROTATION

• Rotations about points other than origin is now
  possible
• 3 step process to accomplish arbitrary point
  rotations
• - Translate to the origin from given point
• - Apply rotation transformation
• - Translate back to the given point

16
ROTATION ABOUT POINT

• Get disired center of rotation to the origin
  (usually negate coordinates page 44)
• Apply rotation matrix to rotate about origin
• Reverse translation back to the point
• 2D rotations about each axis are given in figure
  2-10. Operators and diagrams.

17

• Scaling is controlled by the magnitude of the two
  terms in the primary DIAGONAL of the T matrix
• Using 2 0
• 0 2 a 2 times magnification about
  origin occurs
• Unequal scale factors create distorsion
• Plane surfaces can be easily transformed

18
Combined Transformations

• Typically more than one transform used
• Controls shape and position
• NON-COMMUTATIVE
• Order is important in matrix multiplications
• Consider a 90-degree rotation and then reflection
  on one of the vertices x y of a triangle

19
Combined Transformations

• First rotate and then apply reflection
• Result y x
• Now first apply reflection and then rotate
• Result -y -x
• NOT THE SAME. NON-COMMUTATIVE
• Mapping from one plane into a second plane