This document discusses 2D transformations including translation, rotation, scaling, shearing, and reflection. It explains how to represent points in 2D using vectors and matrices. Various transformation matrices are defined to transform points and geometries through translation, rotation about a pivot point, uniform and non-uniform scaling, reflection across lines or planes, and shearing. Composite transformations consisting of multiple simple transformations applied sequentially are also discussed. Examples are provided to demonstrate how common geometries like lines and polygons are transformed.

1. 2-D Transformations
• Points and lines that joins them, with appropriate drawing
algorithms, represents objects/information graphically.
• Ability to transform these points and lines is basic to
computer graphics.
• In 2-D Objects, transformation is to change the objects:
- Position (Translation)
- Size (Scaling)
- Orientation (Rotation)
- Shapes (Shear)
- Reflection (Mirror image)

2. Point Representation
• A Point is represented in 2-Dimensions by its coordinates.
• We can use a column vector (a 2x1 matrix) or row vector(1x2)
to represent a 2D point : or [x y]
• Row matrices or column matrices are frequently called
position vectors.
• Series of points, each of which is position vectors relative to
some coordinate system is stored in computer as a matrix or
array of numbers.
• Lines are drawn between the points to generate lines, curves
or pictures.

3. Transformation of Points
• [X][T] = [x y] [ ]= [ (ax + cy) (bx + dy)] = [x’ y’]
Where, x’ = (ax + cy) and y’ = (bx + dy)
Case 1: Let, a=d=1 & c=b=0. The transformation matrix [T], then
reduces to the identity matrix. Thus,
[X][T] = [x y] [ ]= [x y] = [x’ y’],no changes as multiplying
with identity matrix gives us same result.
Case 2: Let, d=1, b=c=0,then,
[X][T] = [x y] [ ]= [ax y] = [x’ y’],
where x’ = ax, produces a scale change in x component.
a b
c d
1 0
0 1
a 0
0 1

4. Case 3: Let, b=c=0, i.e.,
[X][T] = [x y] [ ]= [ax dy] = [x’ y’], resulting in scaling
of both x and y components.
- If a ≠ d, means scaling are not equal.
- If a = d > 1, pure enlargement/scaling of coordinates occurs.
- If 0 < a = d < 1, compression of coordinates of point P occurs.
a 0
0 d

5. Case 4: If a and/or d are negative, then reflection through a
plane or axis can be seen. Let b=c=0, d=1,a= -1, then
[X][T] = [x y] [ ]= [-x y] = [x’ y’], reflection through y-
axis results.
- If b=c=0, a=1,d= -1,reflection through x-axis.
- If b=c=0,a=d<0, reflection through origin occurs.
-1 0
0 1

6. Case 5: let a=d=1,c=0,thus
[X][T] = [x y] [ ]= [x (bx+y)] = [x’ y’], it causes
shearing effect. Here x’ is unchanged, but y’ depends on y.
- if a=d=1,b=0. produces shear proportional to y coordinate.
1 b
0 1

7. Straight line transformation
• Position vector are: [A]=[0 1], [B]=[2 3]
• Let transformation matrix, [T] = [ ],due to shearing
effect.
1 2
3 1

8. • Transforming the position vectors for A & B using [T] produces new
vectors A’ and B’ given by:
[A] [T]= [0 1] [ ] = [3 1] = [A’] , and
[B] [T]= [2 3] [ ] = [11 7] = [B’]
• Resulting coordinates for new point A’ are x’=3, y’=1 and B’ are
x’= 11 and y’=7.
• From the Fig., Shearing transformation [T] increased the length of
the line and changes its orientation.
1 2
3 1
1 2
3 1

9. Midpoint Transformation

11. Transformation of Parallel lines
• If a 2x2 matrix is used to transform a pair of parallel lines, the
result is a second pair of parallel lines

12. Transformation of Intersecting lines
• When two intersecting lines are transformed using general
2x2 transformation matrix the resulting lines are also
intersecting. Consider 2 intersection lines
• Points of intersection:
• Transformed point of intersection:

14. Rotation
• 2D Rotation
– Rotation axis
– Rotation angle
– rotation point or pivot point (xr,yr)

15. • 2D Rotation
– If θ is positive  counterclockwise rotation
– If θ is negative  clockwise rotation
– Remember:
• cos(a + b) = cos a cos b - sin a sin b
• cos(a - b) = cos a sin b + sin a cos b
– At first, suppose the pivot point is at the origin
– x’=r cos(θ+Φ) = r cos θ cos Φ - r sin θ sin Φ
y’=r sin(θ+Φ) = r cos θ sin Φ + r sin θ cos Φ
– Let, x = r cos Φ, y = r sin Φ
– x’=x cos θ - y sin θ
y’=x sin θ + y cos θ

16. • 2D Rotation
– P’=R·P, where P is the point with coordinate (x,y) and P’ is
the new point with coordinate (x’,y’).












cos
sin
sin
cos
R
Φ
(x,y)
r
r θ
(x’,y’)

17. • 2D Rotation
– Rotation of a point about any specified position (xr,yr)
x’=xr+(x - xr) cos θ – (y - yr) sin θ
y’=yr+(x - xr) sin θ + (y - yr) cos θ
- Rotations also move objects without deformation
– A line is rotated by applying the rotation formula to each
of the endpoints and redrawing the line between the new
end points
– A polygon is rotated by applying the rotation formula to
each of the vertices and redrawing the polygon using new
vertex coordinates

18. • 2D Inverse Rotation Matrix
• And also:

















1
0
0
0
cos
sin
0
sin
cos
1
R
I
R
R 

*
1

19. Reflection
• Reflection
– Image is generated relative to an axis of reflection by
rotating the object 180° about the reflection axisect
– Transformation that produces a mirror image of an object
– Reflection about the line y=0 (the x axis)
[T] =











1
0
0
0
1
0
0
0
1

20. • Reflection about the line x=0 (the y axis)
[T] =
• Reflection about the origin
[T] =










1
0
0
0
1
0
0
0
1

21. • Reflection about the line y=x
[T] =

22. • Reflection about the line y=-x
[T] =

23. Scaling
• 2D Scaling
– Scaling is used to alter the size of an object
– Simple 2D scaling is performed by multiplying object
positions (x, y) by scaling factors sx and sy
x’ = x · sx
y’ = y · sy
• or P’ = S·P




















y
x
s
s
y
x
y
x
0
0
'
'

24. • 2D Scaling
– Any positive value can be used as scaling factor
• Values less than 1 reduce the size of the object
• Values greater than 1 enlarge the object
• If scaling factor is 1 then the object stays unchanged
• If sx = sy , we call it uniform scaling
• If scaling factor <1, then the object moves closer to the
origin and If scaling factor >1, then the object moves
farther from the origin

25. • 2D Scaling
– We can control the location of the scaled object by
choosing a position called the fixed point (xf,yf)
x’ – xf = (x – xf) sx y’ – yf = (y – yf) sy
x’=x · sx + xf (1 – sx)
y’=y · sy + yf (1 – sy)
– Polygons are scaled by applying the above formula to each
vertex, then regenerating the polygon using the
transformed vertices

26. Composite transformation
• We can setup a sequence of transformations as a composite
transformation matrix by calculating the product of the
individual transformations
• P’=M2·M1·P
=M·P
• Composite 2D Translations
– If two successive translation are applied to a point P, then
the final transformed location P' is calculated as

27. • Composite 2D rotation:
• Composite 2D Scaling:

29. Area scaling example