1. The document discusses various 2D and 3D geometric transformations including translation, rotation, scaling, and their properties. 2. It explains coordinate systems used to specify geometry including world, user, and display coordinates. 3. Key geometric transformations are defined through transformation matrices including translation, rotation, and scaling matrices. Concatenating transformations allows combining multiple transformations.

1. Part - #2
Geometric
Transformation (2D & 3D)
March, 2019

2. 10/5/2022 2
Intro & General Information
Geometric Transformations
Construction
(translate, rotate, scale, mirror)
Viewing
(projections, zooming)
Animation
(processes, vibration)

3. Coordinate Systems
 In order to specify the geometry of a given solid, it is necessary to
use a variety of coordinate systems. Its Major classifications are:
 World Coordinate System: Also known as the "universe" or
sometimes "model" coordinate system. This is the base reference
system for the overall model, ( generally in 3D ), to which all other
model coordinates relate
 User Coordinate System: Also known as “working” coordinate
system. When it is difficult to define certain geometries using
WCS, In such cases user coordinate system can be defined relative
to the WCS.
 Display Coordinates: This refers to the actual coordinates to be
used for displaying the image on the screen.

4. 10/5/2022 4
General Information
Transformation of a point is basic in GT. It can be formulated as follows:
Given a point P that belongs to a geometric model find the corresponding
point P* in the new position such that
P* = f(P, transformation parameters)
• The transformation parameters should provide ONE-TO-ONE-
MAPPING.
• Multiple transformations can be combined to yield a single
transformation which should have the same effect as the sequential
application of original ones. CONCATENATION /kənˌkatnˈāSH(ə)n/
Equation of P* for graphics hardware should be in matrix notation:
P* = [T]P,
where [T] is the transformation matrix.

5. Geometric Transformations
 Sometimes also called modeling
transformations
 Geometric transformations: Changing an object’s
position (translation), orientation (rotation) or size
(scaling)
 Modeling transformations: Constructing a scene
or hierarchical description of a complex object
 Others transformations: reflection and
shearing operations
5

6. 1. Geometric Transformations
 What is transformation?
> It is the backbone of computer graphics, enabling us to
manipulate the shape, size, and location of the object.
> Changing something to something else via rules.
 Why are they important to graphics?
> It can be used to effect the following changes in a geometric
object:
1. Change the location
2. Change the shape
3. Change the size
4. Rotate

7. 2. Types of Transformations
i. Geometric/Object Transformation
> This transformation alters the coordinate values of the object. Basic
operations are scaling, translation, rotation and combination of one
or more of these basic transformations.
> Object transformation = Move (transform) an object in the 2D/3D space.
ii. Visual/ Coordinate System Transformation
> In this transformation there is no change in either the geometry or the
coordinates of the object. A copy of the object is placed at the desired sight,
without changing the coordinate values of the object.
> Coordinate system transformation = Move (transform) the coordinate
system. View the objects from the new coordinate system.

8. 8
3. Types of Geometric Transformations
 Translation
 Rotation
 Scaling
 Reflection/Mirroring.

9. 10/5/2022 9
Translation
Translation is a rigid-body transformation (Euclidean) when
each entity of the model remains parallel, or each point
moves an equal distance in a given direction:
P* = P + d (for both 2D and 3D). In a scalar form (for 3D):
x* = x + xd
y* = y + yd
z* = z + zd
C (11,7)
A (8,5)
B (10,8)
A*
C*
B*
y
x
O
Question: Find the coordinates of vertices A*, B*,
and C* of the translated triangle.
The distance vector of translation: D = [-7 -4]T.
Verify that the lengths of the edges are unchanged.

10. i. Translation - 2D
Moving an object is called a translation.
We translate a point by adding to the x and y coordinates,
respectively, the amount the point should be shifted in the x and y
directions.
We translate an object by translating each vertex in the object.

11. Basic 2D Geometric
Transformations
 2D Translation
 x’ = x + tx , y’ = y + ty
 P’=P+T
 Translation moves the object without deformation
(rigid-body transformation)
P
P’
T
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P ,
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,
11

12. Basic 2D Geometric
Transformations (cont.)
 2D Translation
 To move a line segment, apply the transformation
equation to each of the two line endpoints and
redraw the line between new endpoints
 To move a polygon, apply the transformation
equation to coordinates of each vertex and
regenerate the polygon using the new set of vertex
coordinates
12

13. Scaling is used to change the size of an entity or a model.
P* = [S]P
sx 0 0
For general case [S] = 0 sy 0 ,
0 0 sz
If 0 < s < 1 - compression
If s > 1 - stretching
sx = sy = sz - uniform scaling, otherwise - non-uniform
10/5/2022 13
Scaling
where sx, sy, and sz are the scaling
factors in the X, Y, and Z directions
respectively.
Question: The larger circle is the scaled copy
of the smaller one. Can you say that we have a
uniform scaling? Why? Define y* and R*.
O(4,2)
O*(10,y*)
O
x
y
R 1
R*

14. ii. Scaling
> Changing the size of an object is called a scale. We scale an
object by scaling the x and y coordinates of each vertex in the
object.

15.  2D Rotation
 Rotation axis
 Rotation angle
 rotation point or pivot point (xr,yr)
yr
xr
θ
Basic 2D Geometric
Transformations (cont.)
15
2D Geometric Transformations

16.  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
Basic 2D Geometric
Transformations (cont.)
16
2D Geometric Transformations

17. 10/5/2022 17
Rotation
Rotation is a non-commutative transformation (depends on sequence).
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Question: Let the length of a major and minor
axes of an ellipse with the center on the origin
of the CS be 2a and 2b respectively, and  - the
angle between the major axis and the x-axis.
Then, derive the expression of an ellipse in the
(O,x,y) system.

18. iii. Rotation about the origin
ø Consider rotation about the origin by Θ degrees
ø Radius stays the same, angle increases by Θ.

19.  2D Rotation
 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
 x = r cos Φ, y = r sin Φ
 x’=x cos θ - y sin θ
y’=x sin θ + y cos θ
Φ
(x,y)
r
r θ
(x’,y’)
Basic 2D Geometric
Transformations (cont.)
19
2D Geometric Transformations

20. Basic 2D Geometric
Transformations
 2D Rotation
 P’=R·P
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
cos
sin
sin
cos
R
Φ
(x,y)
r
r θ
(x’,y’)
20
2D Geometric Transformations

21. 3. Transformations as Matrices

22. Cont’d…
Example:
• Translate the rectangle (2, 2), (2, 8), (10, 8), (10, 2)
2 units along x-axis and 3 units along y-axis.

23. Cont’d…

24.  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 · sx
or P’ = S·P
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x
s
s
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x
y
x
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Basic 2D Geometric
Transformations (cont.)
24

25.  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
x’ x
Basic 2D Geometric
Transformations (cont.)
25

26.  2D Scaling
 Why does scaling also reposition object?
 Answer: See the matrix (multiplication)
 Still no clue?

𝑥′
𝑦′
=
𝑠𝑥 0
0 𝑠𝑦
∗
𝑥
𝑦 =
𝑥 ∗ 𝑠𝑥 + 𝑦 ∗ 0
𝑥 ∗ 0 + 𝑦 ∗ 𝑠𝑦
Basic 2D Geometric
Transformations (cont.)
26

27.  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
Basic 2D Geometric
Transformations (cont.)
27

28. Example:
 If the triangle A(1, 1), B(2, 1), C(1, 3) is scaled by a
factor 2, find the new coordinates of the triangle.

29. Cont’d…

30. Cont’d…
Example:
• Rotate the rectangle (0, 0), (2, 0), (2, 2), (0, 2) by 30o
ccw about its centroid and find the new coordinates
of the rectangle.

31. Summary

32. 4. Homogeneous Coordinates
> In order to represent a translation as a matrix multiplication
operation we use 3 x 3 matrices and pad the points to become 3 x 1
matrices.
> This coordinate system (using three values to represent a 2D point) is
called homogeneous coordinates.
> Let the added extra coordinate be W, to a point: P (x, y, W).
> Two sets of homogeneous coordinates represent the same point if
they are a multiple of each other.
> If W ≠ 0 , divide by it to get Cartesian coordinates of point: (x/W,
y/W, 1).
> If W = 0, point is said to be at infinity.

33. Cont’d…
Transformation matrices for 2D translation are now 3x3.

34. 5. Concatenation of transformation

35. Cont’d…

36. Properties of Translations

37. Homogeneous form of scale
Concatenation of scales

39. 10/5/2022 39
Homogeneous Transformation
When we scale then rotate, the transformed image is given by:
P* = ([R][S])P
where [S], [R], [R] [S] are 3x3 transformation matrices. This is not the case
for a translation (P* = P + d). The goal is to find a [D] such that
P + d = [D]P
in order to perform valid matrix multiplication.
This is found by using a homogeneous coordinates.
Homogeneous Transformation maps n-dimensional space into (n+1)- dim.
3D representation of the point vector - P = [x, y, z]T
Homogeneous rep. of the same vector - P = [xw, yw, zw, w]T where w = 1

40. Matrix Representations and
Homogeneous Coordinates
 Many graphics applications involve
sequences of geometric transformations
 Animations
 Design and picture construction applications
 We will now consider matrix representations
of these operations
 Sequences of transformations can be efficiently
processed using matrices
40
2D Geometric Transformations

41. Matrix Representations and
Homogeneous Coordinates (cont.)
 P’ = M1 · P + M2
 P and P’ are column vectors
 M1 is a 2 by 2 array containing multiplicative
factors
 M2 is a 2 element column matrix containing
translational terms
 For translation M1 is the identity matrix
 For rotation or scaling, M2 contains the
translational terms associated with the pivot point
or scaling fixed point
41
2D Geometric Transformations

42.  To produce a sequence of operations, such
as scaling followed by rotation then
translation, we could calculate the
transformed coordinates one step at a time
 A more efficient approach is to combine
transformations, without calculating
intermediate coordinate values
Matrix Representations and
Homogeneous Coordinates (cont.)
42
2D Geometric Transformations

43.  Multiplicative and translational terms for a 2D
geometric transformation can be combined
into a single matrix if we expand the
representations to 3 by 3 matrices
 We can use the third column for translation terms,
and all transformation equations can be
expressed as matrix multiplications
Matrix Representations and
Homogeneous Coordinates (cont.)
43
2D Geometric Transformations

44.  Expand each 2D coordinate (x,y) to three
element representation (xh,yh,h) called
homogeneous coordinates
 h is the homogeneous parameter such that
x = xh/h, y = yh/h,
  infinite homogeneous representations for a
point
 A convenient choice is to choose h = 1
Matrix Representations and
Homogeneous Coordinates (cont.)
44
2D Geometric Transformations

45. 10/5/2022 45
Homogeneous Transformation
The transformation matrices in new (homogeneous) representation:
   
   
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33
32
31
23
22
21
13
12
11
r
r
r
r
r
r
r
r
r
M
s
s
s
S
z
y
x
D
z
y
x
d
d
d

46.  2D Translation Matrix
or, P’ = T(tx,ty)·P
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x
t
t
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y
x
Matrix Representations and
Homogeneous Coordinates (cont.)
46
2D Geometric Transformations

47.  2D Rotation Matrix
or, P’ = R(θ)·P
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cos
1
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y
x
y
x
Matrix Representations and
Homogeneous Coordinates (cont.)
47
2D Geometric Transformations

48.  2D Scaling Matrix
or, P’ = S(sx,sy)·P
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Matrix Representations and
Homogeneous Coordinates (cont.)
48
2D Geometric Transformations

49. Inverse Transformations
 2D Inverse Translation Matrix
 By the way:
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1
y
x
t
t
T
I
T
T 

*
1
49
2D Geometric Transformations

50. Inverse Transformations (cont.)
 2D Inverse Rotation Matrix
 And also:
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0
0
0
cos
sin
0
sin
cos
1
R
I
R
R 

*
1
50
2D Geometric Transformations

51. Inverse Transformations (cont.)
 2D Inverse Rotation Matrix:
 If θ is negative  clockwise
 In
 Only sine function is affected
 Therefore we can say
 Is that true?
 Proof: It’s up to you 
I
R
R 

*
1
T
R
R 
1
51
2D Geometric Transformations

52. Inverse Transformations (cont.)
 2D Inverse Scaling Matrix
 Of course:
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1
y
x
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s
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I
S
S 
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1
52
2D Geometric Transformations

53. 2D Composite Transformations
 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
53
2D Geometric Transformations

54. 2D Composite Transformations
(cont.)
 Composite 2D Translations
 If two successive translation are applied to a point P,
then the final transformed location P' is calculated as
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,
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2 y
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54
2D Geometric Transformations

55.  Composite 2D Rotations
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






































1
0
0
0
)
cos(
)
sin(
0
)
sin(
)
cos(
1
0
0
0
cos
sin
0
sin
cos
1
0
0
0
cos
sin
0
sin
cos
2
1
2
1
2
1
2
1
1
1
1
1
2
2
2
2
P
R
P 

 )
(
' 2
1 

2D Composite Transformations
(cont.)
55
2D Geometric Transformations

56.  Composite 2D Scaling


































1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
2
1
2
1
1
1
2
2
y
y
x
x
y
x
y
x
s
s
s
s
s
s
s
s
)
,
(
)
,
(
)
,
( 2
1
2
1
1
1
2
2 y
y
x
x
y
x
y
x s
s
s
s
s
s
s
s 


 S
S
S
2D Composite Transformations
(cont.)
56
2D Geometric Transformations

57.  Don’t forget:
 Successive translations are additive
 Successive scalings are multiplicative
 For example: If we triple the size of an object
twice, the final size is nine (9) times the original
 9 times?
 Why?
 Proof: Again up to you 
2D Composite Transformations
(cont.)
57
2D Geometric Transformations

58. General Pivot Point Rotation
 Steps:
1. Translate the object so that the pivot point is
moved to the coordinate origin.
2. Rotate the object about the origin.
3. Translate the object so that the pivot point is
returned to its original position.
58
2D Geometric Transformations

59. General Pivot Point Rotation
59
2D Geometric Transformations

60.  General 2D Pivot-Point Rotation







































1
0
0
1
0
0
1
1
0
0
0
cos
sin
0
sin
cos
1
0
0
1
0
0
1
r
r
r
r
y
x
y
x
























1
0
0
sin
)
cos
1
(
cos
sin
sin
)
cos
1
(
sin
cos
r
r
r
r
x
y
y
x
2D Composite Transformations
(cont.)
60
2D Geometric Transformations

61. General Fixed Point Scaling
 Steps:
1. Translate the object so that the fixed point
coincides with the coordinate origin.
2. Scale the object about the origin.
3. Translate the object so that the pivot point is
returned to its original position.
61
2D Geometric Transformations

62. General Fixed Point Scaling
(cont.)
(xr, yr) (xr, yr)
62
2D Geometric Transformations

63. 1 0 𝑥𝑓
0 1 𝑦𝑓
0 0 1
.
𝑠𝑥 0 0
0 𝑠𝑦 0
0 0 1
.
1 0 − 𝑥𝑓
0 1 − 𝑦𝑓
0 0 1
=
𝑠𝑥 0 𝑥𝑓(1 − 𝑠𝑥)
0 𝑠𝑦 𝑦𝑓(1 − 𝑠𝑦)
0 0 1
𝐓(𝑥𝑓, 𝑦𝑓) ⋅ 𝐒(𝑠𝑥, 𝑠𝑦) ⋅ 𝐓(−𝑥𝑓, −𝑦𝑓) = 𝐒(𝑥𝑓, 𝑦𝑓, 𝑠𝑥, 𝑠𝑦)
• General 2D Fixed-Point Scaling:
General Fixed Point Scaling
(cont.)
63
2D Geometric Transformations

64. 2D Composite Transformations
(cont.)
 General 2D scaling directions:
 Above: scaling parameters were along x and y
directions
 What about arbitrary directions?
 Answer: See next slides
64
2D Geometric Transformations

65. General 2D Scaling Directions
Scaling parameters s1 and s2 along orthogonal directions defined by the
angular displacement θ. 65
2D Geometric Transformations

66. General 2D Scaling Directions
(cont.)
 General procedure:
1. Rotate so that directions coincides with x and y
axes
2. Apply scaling transformation 𝑆 𝑠1, 𝑠2
3. Rotate back
 The composite matrix:
𝑅−1 Θ ∗ 𝑆 𝑠1, 𝑠2 ∗ 𝑅 Θ =
𝑠1 cos2 Θ + 𝑠2 sin2 Θ 𝑠2 − 𝑠1 cos Θ sin Θ 0
𝑠2 − 𝑠1 cos Θ sin Θ 𝑠1 sin2 Θ + 𝑠2 cos2 Θ 0
0 0 1
66
2D Geometric Transformations

67. 2D Composite Transformations
(cont.)
 Matrix Concatenation Properties:
 Matrix multiplication is associative !
 M3· M2· M1= (M3· M2 ) · M1 = M3· ( M2 · M1 )
 A composite matrix can be created by multiplicating left-
to-right (premultiplication) or right-to-left
(postmultiplication)
 Matrix multiplication is not commutative !
 M2 · M1 ≠ M1 · M2
67
2D Geometric Transformations

68. 2D Composite Transformations
(cont.)
 Matrix Concatenation Properties:
 But:
 Two successive rotations
 Two successive translations
 Two successive scalings
 are commutative!
 Why?
 Proof: You got it: Up to you  
68
2D Geometric Transformations

69. 10/5/2022 6
9
Mirror
Plane* => Negate the corresponding coordinate
Mirror through Line* => Reflect through 2 planes intersecting at the axis
Point* => Reflect through 3 planes intersecting at the point
* plane - principal plane, line - X, Y, or Z axes, point - CS origin
P* = [M]P,
where [M] = =
Question: Define the signs (in the matrix)
for the reflections (mirroring) through:
a) x = 0, y = 0, z = 0 planes
b) X, Y, and Z axes
c) the CS origin
33
22
11
0
0
0
0
0
0
m
m
m
1
0
0
0
1
0
0
0
1




70. Reversing the order
in which a sequence of transformations is performed may affect the transformed
position of an object.
In (a), an object is first translated in the x direction, then rotated counterclockwise
through an angle of 45°.
In (b), the object is first rotated 45° counterclockwise, then translated in the x
direction
70
2D Geometric Transformations

71. Other 2D Transformations
 Reflection
 Transformation that produces a mirror image of an
object
71
2D Geometric Transformations

72.  Reflection
 Image is generated relative to an axis of reflection
by rotating the object 180° about the reflection
axis
 Reflection about the line y=0 (the x axis) (previous
slide)











1
0
0
0
1
0
0
0
1
Other 2D Transformations (cont.)
72
2D Geometric Transformations

73. Other 2D Transformations (cont.)
 Reflection
 Reflection about the line x=0 (the y axis)










1
0
0
0
1
0
0
0
1
73
2D Geometric Transformations

74. −1 0 0
0 − 1 0
0 0 1
 Reflection about the origin
Other 2D Transformations (cont.)
74
2D Geometric Transformations

75.  Reflection about the line y=x
0 1 0
1 0 0
0 0 1
Other 2D Transformations (cont.)
75
2D Geometric Transformations

76.  Reflection about the line y=-x
0 − 1 0
−1 0 0
0 0 1
Other 2D Transformations (cont.)
76
2D Geometric Transformations

77.  Shear
 Transformation that distorts the shape of an object
such that the transformed shape appears as the
object was composed of internal layers that had
been caused to slide over each other
y
x
(0,1) (1,1)
(1,0)
(0,0)
y
x
(2,1) (3,1)
(1,0)
(0,0)
shx=2
Other 2D Transformations (cont.)
77
2D Geometric Transformations

78.  Shear
 An x-direction shear relative to the x axis
 An y-direction shear relative to the y axis










1
0
0
0
1
0
0
1
y
sh










1
0
0
0
1
0
0
1 x
sh
y
y
y
sh
x
x x




'
'
Other 2D Transformations (cont.)
78
2D Geometric Transformations

79.  Shear
 x-direction shear relative to other reference lines









 
1
0
0
0
1
0
*
1 ref
x
x y
sh
sh
 
y
y
y
y
sh
x
x ref
x




'
*
'
Other 2D Transformations (cont.)
79
2D Geometric Transformations

80. Example
A unit square (a) is transformed to a shifted parallelogram
(b) with shx = 0.5 and yref = −1 in the shear matrix from Slide 56
80
2D Geometric Transformations

81.  Shear
 y-direction shear relative to the line x = xref











1
0
0
*
1
0
0
1
ref
y
y x
sh
sh
 
ref
y x
x
sh
x
y
x
x




*
'
'
Other 2D Transformations (cont.)
81
2D Geometric Transformations

82. Example
A unit square (a) is turned into a shifted parallelogram
(b) with parameter values shy = 0.5 and xref = −1 in the y -direction shearing
transformation from Slide 58
82
2D Geometric Transformations

83.  This slide is intentionally left blank
 Your responsibility to fill it 
Raster Methods for
Transformations and OpenGL
83
2D Geometric Transformations

84. Transformation Between
Coordinate Systems
 Individual objects may be defined in their
local cartesian reference system.
 The local coordinates must be transformed
to position the objects within the scene
coordinate system.
84
2D Geometric Transformations

85. Steps for coordinate transformation
1. Translate so that the origin (x0, y0 ) of the
x′-y′ system is moved to the origin of the
x-y system.
2.Rotate the x′ axis on to the axis x.
Transformation Between
Coordinate Systems
85
2D Geometric Transformations

86. y
x
θ
x0
y0
0
Transformation Between
Coordinate Systems (cont.)
86
2D Geometric Transformations

87. y
x
x0
y0
0
θ
Transformation Between
Coordinate Systems (cont.)
87
2D Geometric Transformations

88. y
x
x′ x0
y0
0
y′
Transformation Between
Coordinate Systems (cont.)
88
2D Geometric Transformations

89. 𝐓(−𝑥0, −𝑦0) =
1 0 − 𝑥0
0 1 − 𝑦0
0 0 1
𝐑(−𝜃) =
𝐶𝑜𝑠𝜃 𝑆𝑖𝑛𝜃 0
−𝑆𝑖𝑛𝜃 𝐶𝑜𝑠𝜃 0
0 0 1
𝐌𝑥𝑦,𝑥′𝑦′ = 𝐑(−𝜃) ⋅ 𝐓(−𝑥0, −𝑦0)
Transformation Between
Coordinate Systems (cont.)
89
2D Geometric Transformations

90. An alternative method:
-Specify a vector V that indicates the direction
for the positive y′ axis. Let
-Obtain the unit vector u=(ux ,u y) along the x′
axis by rotating v 900 clockwise.
𝐯 =
𝐕
𝐕
= (𝑣𝑥, 𝑣𝑦)
Transformation Between
Coordinate Systems (cont.)
90
2D Geometric Transformations

91.  Elements of any rotation matrix can be
expressed as elements of orthogonal unit
vectors. That is, the rotation matrix can be
written as
𝐑 =
𝑢𝑥 𝑢𝑦 0
𝑣𝑥 𝑣𝑦 0
0 0 1
Transformation Between
Coordinate Systems (cont.)
91
2D Geometric Transformations

92. y
x
x0
y0
0
V
Transformation Between
Coordinate Systems (cont.)
92
2D Geometric Transformations

93. OpenGL Geometric
Transformation Functions
 A separate function is available for each of the
basic geometric transformations
AND
 All transformations are specified in three
dimensions
 Why?
 Answer: Remember; OpenGL was developed as
3D library
 But how to perform 2D transformations?
 Answer: Set z = 0
93
2D Geometric Transformations

94. Basic OpenGL Geometric
Transformations
 Translation
 glTranslate* (tx, ty, tz);
 * is either f or d
 tx, ty and tz are any real number
 For 2D, set tz=0.0
 Rotation
 glRotate* (theta, vx, vy, vz);
 * is either f or d
 theta is rotation angle in degrees (internally converted to
radian)
 Vector v=(vx, vy, vz) defines the orientation for a rotation axis
that passes through the coordinate origin
 For 2D, set vz=1.0 and vx=vy=0.0
94
2D Geometric Transformations

95. Basic OpenGL Geometric
Transformations (cont.)
 Scaling
 glScale* (sx, sy, sz);
 * is either f or d
 sx, sy and sz are any real number
 Negative values generate reflection
 Zero values can cause error because inverse matrix
cannot be calculated
 All routines construct a 4x4 transformation
matrix
 OpenGL uses composite matrices
 Be careful with the order 95
2D Geometric Transformations

96. OpenGL Matrix Operations
 glMatrixMode(.);
 Projection Mode: Determines how the scene is
projected onto the screen
 Modelview Mode: Used for storing and combining
geometric transformations
 Texture Mode: Used for mapping texture patterns
to surfaces
 Color Mode: Used to convert from one color mode
to another
96
2D Geometric Transformations

97. OpenGL Matrix Operations
 Modelview matrix, used to store and combine
geometric transformations
 glMatrixMode(GL_MODELVIEW);
 A call to a transformation routine generates a
matrix that is multiplied by the current matrix
 To assign the identity matrix to the current
matrix
 glLoadIdentity();
97
2D Geometric Transformations

98. OpenGL Matrix Operations
(cont.)
 Alternatively:
 glLoadMatrix* (elements16);
 To assign other values to the elements of the
current matrix
 In column-major order:
 First four elements in first column
 Second four elements in second column
 Third four elements in third column
 Fourth four elements in fourth column
98
2D Geometric Transformations

99. OpenGL Matrix Operations
(cont.)
 Concatenating a specified matrix with current
matrix:
 glMultMatrix* (otherElements16);
 Current matrix is postmultiplied (right-to-left) by
the specified matrix
 Warning:
 Matrix notation mjk means:
 In OpenGL: j  column, k  row
 In mathematics: j  row, k  column
99
2D Geometric Transformations

100. OpenGL Matrix Stacks
 OpenGL maintains a matrix stack for
transformations
 Initially the modelview stack contains only the
identity matrix
 More about it:
 Coming soon
100
2D Geometric Transformations

101. OpenGL Transformation Routines
 For example, assume we want to do in the following
order:
 translate by +2, -3, +4,
 rotate by 450 around axis formed between origin and 1, 1, 1
 scale with respect to the origin by 2 in each direction.
 Our code would be
glMatrixMode(GL_MODELVIEW);
glLoadIdentity(); //start with identity
glScalef(2.0,2.0,2.0); //Note: Start with the LAST operation
glRotatef(45.0,1.0,1.0,1.0);
glTranslatef(2.0,-3.0, 4.0); //End with the FIRST operation
101
2D Geometric Transformations

102. OpenGL Transformation Functions
102
2D Geometric Transformations

103. Next Lecture
3D Geometric Transformations
103

104. References
 Donald Hearn, M. Pauline Baker, Warren R.
Carithers, “Computer Graphics with OpenGL, 4th
Edition”; Pearson, 2011
 Sumanta Guha, “Computer Graphics Through
OpenGL: From Theory to Experiments”, CRC Press,
2010
 Edward Angel, “Interactive Computer Graphics. A
Top-Down Approach Using OpenGL”, Addison-
Wesley, 2005
104

105. Ex. 1: Rotation of object about arbitrary point P (x, y) by θ
> Translate point (x, y) to origin
by (-x, -y)
> Rotate by θ
> Translate the point by (x, y)

106. 6. Reflections/Mirroring

107. More Reflections

108. 3 D - Rigid body transformations
 Methods for object modeling transformation in three dimensions
are extended from two dimensional methods by including
consideration for the z coordinate.
 Preserve lines, angles and distances.
# Generalize from 2D by including z
coordinate
# Straight forward for translation and
scale
# Rotation more difficult
# Homogeneous coordinates: 4
components
# Transformation matrices: 4×4
éléments.

109. 3D Point:
 We will consider points as column vectors.
 Thus, a typical point with coordinates (x, y, z) is
represented as:
3D Point Homogenous Coordinate:
 A 3D point P is represented in homogeneous
coordinates by a 4-dim, Vector.
 We don't lose anything.
 The main advantage: it is easier to compose
translation and rotation.
 Everything is matrix multiplication.

110. i. 3D Translation
 In homogeneous coordinates, 3D transformations are represented by
4×4 matrixes:.
 P is translated to P' by:
 An object is translated in 3D
dimensional by transforming
each of the defining points of
the objects .

111. Cont’d…
# An object represented as a set of polygon surfaces, is translated by
translate each vertex of each surface and redraw the polygon
facets in the new position.

112. ii. 3D Rotation
 In general, rotations are specified by a rotation axis and an angle.
 In two-dimensions there is only one choice of a rotation axis that leaves
points in the plane.
 The easiest rotation axes are those that parallel to the coordinate axis.
 Positive rotation angles produce counterclockwise rotations about a
coordinate axis, if we are looking along the positive half of the axis
toward the coordinate origin.

113. Rotation about Z-axis:
:For z axis same as 2D
rotation:
P′ = Rz(θ) ⋅P
Rotation about X-axis:
P′ = Rx(θ) ⋅P

114. Cont’d…
• Y-axis rotation:
X’ cos(θ) 0 -sin(θ) 0
Y’ = 0 1 0 0
Z’ sin(θ) 0 cos(θ) 0
1 0 0 0 1

115. General 3D Rotations about axis // to Coordinate Axis
 Rotation axis parallel with coordinate axis (Example x axis):

116. iii. 3D Scaling
# About origin: changes the size of the object and repositions the
object relative to the coordinate original.
P′ = S ⋅ P

117. • Scale about any arbitrary point:

118. iV. 3D Reflections
 About an axis: equivalent to 180°rotation about that axis.
 About a plane: A reflection through the xy plane.
 A reflections through the xz and the yz planes are defined
similarly.

119. Exercise
 You sat in the car, and find the side mirror 40mm on your
right and 30mm in your front. You started your car and drove
200mm forward, turned 30 degrees to right, moved 200mm
forward again, and turned 45 degrees to the right, and
stopped.
 What is the position of the side mirror now, relative to where
you were sitting in the beginning?

120. Solution
# The side mirror position is locally (40, 30).
# The matrix of first driving forward 200mm is:

121. Cont’d…
 The matrix to turn to the right 30 deg and 45 degrees
(rotating -30 and -45 degrees about the origin) are:
The 3rd transformation is moving
by 200mm along an axis oriented 60
degree with the x-axis.