1. Lecture 3 – Transformation And Coordinate Systems
Computer Graphics CS630
Lecture 3 – Transformation And Coordinate Systems

2. What is a Transformation?
Maps points (x, y) in one coordinate system to points (x', y') in another coordinate system
x' = ax + by + c
y' = dx + ey + f

3. Transformations
Simple transformation
Translation
Rotation
Scaling

4. Transformations Deformable transformations Issues Shearing Tapering
Twisting
Etc..
Issues
Can be combined
Are these operations invertible?

5. Transformations Why use transformations?
Position objects in a scene (modeling)
Change the shape of objects
Create multiple copies of objects
Projection for virtual cameras
Animations

6. Classes of Transformation
Rigid-Body/ Euclidean transformation
Similarity Transforms
Linear Transforms
Affine Transforms
Projective Transforms

7. Rigid-Body / Euclidean Transforms
Preserves distances
Preserves angles
Rigid / Euclidean
Identity
Translation
Rotation

8. How are Transforms Represented?
x' = ax + by + c
y' = dx + ey + f x' y'
a b
d e x y c f = +
p' = M p t

9. Translation = + = +

10. Properties of Translation= = = =

11. Scaling
Uniform scaling iff

12. Rotations (2D)

13. Rotations 2D
So in matrix notation

14. Rotations (3D)

15. Properties of Rotations
order matters!

16. Combining Translation & Rotation

17. Combining Translation & Rotation

18. Homogeneous Coordinates
Add an extra dimension
in 2D, we use 3 x 3 matrices
In 3D, we use 4 x 4 matrices
Each point has an extra value, w x' y' z' w' a e i m b f j n c g k o d h l p x y z w =
p' = M p

19. Homogeneous Coordinates
Most of the time w = 1, and we can ignore it x' y' z' 1 a e i b f j c g k d h l 1 x y z 1 =

20. Homogeneous Coordinates
can be represented as
where

21. Translation Revisited

22. Rotation & Scaling Revisited

23. Combining Transformations
where

24. Transforming Tangents

25. Transforming Normals

26. Surface Normal
Surface Normal: unit vector that is locally perpendicular to the surface

27. Why is the Normal important?
It's used for shading — makes things look 3D!
object color only
Diffuse Shading

28. Visualization of Surface Normal
± x = Red ± y = Green ± z = Blue

29. How do we transform normals?
nWS
nOS
World Space
Object Space

30. Transform the Normal like the Ray?
translation?
rotation?
isotropic scale?
scale?
reflection?
shear?
perspective?

31. More Normal Visualizations
Incorrect Normal Transformation
Correct Normal Transformation

32. Transforming Normals

33. Rotations about an arbitrary axis
Rotate by around a unit axis

34. An Alternative View
We can view the rotation around an arbitrary axis as a set of simpler steps
We know how to rotate and translate around the world coordinate system
Can we use this knowledge to perform the rotation?

35. Rotation about an arbitrary axis
Translate the space so that the origin of the unit vector is on the world origin
Rotate such that the extremity of the vector now lies in the xz plane (x-axis rotation)
Rotate such that the point lies in the z-axis (y-axis rotation)
Perform the rotation around the z-axis
Undo the previous transformations

36. Rotation about an arbitrary axis
Step 1
Rotate x-axis y
(a,b,c) x x
(a’,b’,c’) z

37. Closer Look at Y-Z Plane
Need to rotate  degrees around the x-axis y  z

38. Equations for 

39. Rotation about the Y-axis
Using the same analysis as before, we need to rotate  degrees around the Y-axis y x
(a’,b’,c’)=Rx () (a,b,c)T z

40. Equations for 

41. Rotation about the Z-axis
Now, it is aligned with the Z-axis, thus we can simply rotate  degrees around the Z-axis.
Then undo all the transformations we just did

42. Equation summary

43. Deformations Transformations that do not preserve shape
Non-uniform scaling
Shearing
Tapering
Twisting
Bending

44. Shearing

45. Tapering

46. Twisting

47. Bending

48. Quick Recap
Computer Graphics is using a computer to generate an image from a representation.
computer
Model
Image

49. Modeling
What we have been studying so far is the mathematics behind the creation and manipulation of the 3D representation of the object.
computer
Model
Image

50. What have we seen so far? Basic representations (point, vector)
Basic operations on points and vectors (dot product, cross products, etc.)
Transformation – manipulative operators on the basic representation (translate, rotate, deformations) – 4x4 matrices to “encode” all these.

51. Why do we need this?
In order to generate a picture from a model, we need to be able to not only specify a model (representation) but also manipulate the model in order to create more interesting images.

52. Overview
The next set of slides will deal with the other half of the process.
From a model, how do we generate an image
computer
Model
Image

53. Scene Description
Scene
Camera
Lights
Background
Materials
Objects

54. Graphics Pipeline

55. Graphics Pipeline
Modeling transforms orient the models within a common coordinate frame (world space)

56. Graphics Pipeline

57. Graphics Pipeline Maps world space to eye space
Viewing position is transformed to origin & direction is oriented along some axis (usually z)

58. Graphics Pipeline Transform to Normalized Device Coordinates (NDC)
Portions of the object outside the view volume (view frustum) are removed

59. Graphics Pipeline
The objects are projected to the 2D image place (screen space)

60. Graphics Pipeline

61. Graphics Pipeline
Z-buffer - Each pixel remembers the closest object (depth buffer)

62. Graphics Pipeline
Almost every step in the graphics pipeline involves a change of coordinate system. Transformations are central to understanding 3D computer graphics.

63. Intuitively World Object Space Space Camera Space +Projection +NDC
Rasterization

64. Coordinate Systems Object coordinates World coordinates
Camera coordinates
Normalized device coordinates
Window coordinates

65. Object Coordinates
Convenient place to model the object

66. World Coordinates
Common coordinates for the scene

67. Positioning Synthetic Camera
What are our “degrees of freedom” in camera positioning?
To achieve effective visual simulation, we want:
1) the eye point to be in proximity of modeled scene
2) the view to be directed toward region of interest, and
3) the image plane to have a reasonable “twist”

68. Eye Coordinates Eyepoint at origin
u axis toward “right” of image plane
v axis toward “top” of image plane
view direction along negative n axis

69. Transformation to Eye Coordinates
Our task: construct the transformation M that re-expresses world coordinates in the viewer frame

70. Machinery: Changing Orthobases
Suppose you are given an orthobasis u, v, n
What is the action of the matrix M with
rows u, v, and n as below?

71. Applying M to u, v, n
Two equally valid interpretations, depending on reference frame:
1: Think of uvn basis as a rigid object in a fixed world space
Then M “rotates” uvn basis into xyz basis
2: Think of a fixed axis triad, with “labels” from xyz space
Then M “reexpresses” an xyz point p in uvn coords!
It is this second interpretation that we use today
to “relabel” world-space geometry with eye space coordinates

72. Positioning Synthetic Camera
Given eyepoint e, basis ˆu, ˆv, ˆn
Deduce M that expresses world in eye coordinates:
Overlay origins, then change bases:

73. Positioning Synthetic Camera
Check: does M re-express world geometry in eye coordinates?

74. Positioning Synthetic Camera
Camera specification must include:
World-space eye position e
World-space “lookat direction” -n
Are e and -n enough to determine the camera DOFs
(degrees of freedom)?

75. Positioning Synthetic Camera
Are e and -n enough to determine the camera DOFs?
No. Note that we were not given u and v!
(Why not simply require the user to specify them?)
We must also determine u and v, i.e., camera “twist” about n.
Typically done by specification of a world-space “up vector”
provided by user interface, e.g., using gluLookat(e, c, up)
“Twist” constraint: Align v with world up vector (How?)

76. Positioning Synthetic Camera

77. Where are we?

78. What is Projection?
Any operation that reduces dimension (e.g., 3D to 2D)
Orthographic Projection
Perspective Projection

79. Orthographic Projection
focal point at infinity
rays are parallel and orthogonal to the image plane F
Image
World I W
Image
World F

80. Comparison

81. Simple Perspective Camera
camera looks along z-axis
focal point is the origin
image plane is parallel to xy-plane at distance d
d is call focal length

82. Similar Triangles Similar situation with x-coordinateY Z
[0, d]
[0, 0]
[Y, Z]
[(d/Z)Y, d]
Similar situation with x-coordinate
Similar Triangles: point [x,y,z] projects to [(d/z)x, (d/z)y, d]

83. Divide by 4th coordinate
Projection Matrix
Projection using homogeneous coordinates:
transform [x, y, z] to [(d/z)x, (d/z)y, d] d 1 é ë ê ù û ú x y z = dx dy dz [ ] Þ
Divide by 4th coordinate
(the “w” coordinate)
2-D image point:
discard third coordinate
apply viewport transformation to obtain physical pixel coordinates

84. Perspective Projection

85. Perspective Projection
z = 0 not allowed (what happens to points on plane z = 0?)
Operation well-defined for all other points

86. Perspective Projection
Matrix formulation using “homogeneous 4-vectors”:
Finally, recover projected point using homogenous convention:
Divide by 4th element to convert 4-vector to 3-vector:

87. Camera Coordinates
Coordinate system with the camera in a convenient pose y x z

88. View Volume and Normalized Device Coordinates

89. Normalized Device Coordinates
Device independent coordinates
Visible coordinate usually range from:

90. Perspective Projection
Taking the camera coordinates to NDC

91. Perspective Projection

92. Perspective Projection + NDC

93. Window Coordinates Adjusting the NDC to fit the window
is the lower left of the window

94. Window Coordinates Adjusting the NDC to fit the window
is the lower left of the window
height
width

95. Window Coordinates

96. Summary : Object Coordinate to Device Coordinate
Take your representation (points) and transform it from Object Space to World Space (Mwo)
Take your World Space point and transform it to Camera Space (Mcw)
Perform the remapping and projection onto the image plane in Normalized Device Coordinates (Mw_p Mpc)
Perform this set of transformations on each point of the polygonal object (M= Mw_pMpcMcwMwo)