0. CSCE 441 Computer Graphics 3-D Viewing
Jinxiang Chai

1. Outline 3D Viewing Required readings: HB 10-1 to 10-10
Compile and run the codes in page 388
- opengl three-dimensional viewing program example 1

2. Taking Pictures Using A Real Camera
Steps:
- Identify interesting objects
- Rotate and translate the camera to a desired camera viewpoint
- Adjust camera settings such as focal length
- Choose desired resolution and aspect ratio, etc.
- Take a snapshot

3. Taking Pictures Using A Real Camera
Steps:
- Identify interesting objects
- Rotate and translate the camera to a desired camera viewpoint
- Adjust camera settings such as focal length
- Choose desired resolution and aspect ratio, etc.
- Take a snapshot
Graphics does the same thing for rendering an image for 3D geometric objects

4. Rendering Images Using a Virtual Camera

5. 3D Geometry Pipeline Object space World space View space
Rotate and translate the camera
Object space
World space
View space
Focal length
Aspect ratio & resolution
Normalized projection space
Screen/Image space 5

6. 3D Geometry Pipeline Model space (Object space)
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
Model space
(Object space)

7. 3D Geometry Pipeline World space (Object space)
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
World space
(Object space)

8. 3D Geometry Pipeline Eye space (View space)
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
Eye space
(View space)

9. Normalized projection space
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
Normalized projection space

10. Image space, window space, raster space, screen space, device space
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
Image space, window space, raster space, screen space, device space

11. Normalized project space
Taking Steps Together C M
Object space
World space
View space P V
Normalized project space
Image space 11

12. OpenGL Codes 12

13. OpenGL Codes 13

14. OpenGL Codes 14

15. Normalized project space
3D Geometry Pipeline
Object space
World space
View space
Normalized project space
Image space

16. Translate, scale &rotate
3D Geometry Pipeline
Translate, scale &rotate
Object space
World space
glTranslate*(tx,ty,tz)

17. Translate, scale &rotate
3D Geometry Pipeline
Translate, scale &rotate
Object space
World space
glScale*(sx,sy,sz)

18. Translate, scale &rotate
3D Geometry Pipeline
Translate, scale &rotate
Object space
World space
Rotate about r by the angle
glRotate*

19. Normalized projection space
3D Geometry Pipeline
Object space
World space
View space
Normalized projection space
Image space
Screen space

20. 3D Geometry Pipeline World space View space
Now look at how we would compute the world->eye transformation
World space
View space

21. 3D Geometry Pipeline World space View space
Now look at how we would compute the world->eye transformation
Rotate&translate
World space
View space

22. Camera Coordinate

23. Viewing Trans: gluLookAt
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz) 23

24. Camera Coordinate Canonical coordinate system
- usually right handed (looking down –z axis)
- convenient for project and clipping

25. Camera Coordinate Mapping from world to eye coordinates
- eye position maps to origin
- right vector maps to x axis
- up vector maps to y axis
- back vector maps to z axis

26. Viewing Transformation
We have the camera in world coordinates
We want to model transformation T which takes object from world to camera

27. Viewing Transformation
We have the camera in world coordinates
We want to model transformation T which takes object from world to camera
Trick: find T-1 taking object from camera to world

28. Viewing Transformation
We have the camera in world coordinates
We want to model transformation T which takes object from world to camera
Trick: find T-1 taking object from camera to world ?

29. Review: 3D Coordinate Trans.
Transform object description from to p

30. Review: 3D Coordinate Trans.
Transform object description from to p 30

31. Review: 3D Coordinate Trans.
Transform object description from camera to world

32. Viewing Transformation
Trick: find T-1 taking object from camera to world
- eye position maps to origin
- back vector maps to z axis
- up vector maps to y axis
- right vector maps to x axis

33. Viewing Transformation
Trick: find T-1 taking object from camera to world
H&B equation (10-4)

34. Viewing Trans: gluLookAt
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

35. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
How to determine ?

36. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
How to determine ?

37. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

38. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

39. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Stop here.
H&B equation (10-1)

40. Viewing Trans: gluLookAt
Test: gluLookAt( 4.0, 2.0, 1.0, 2.0, 4.0, -3.0, 0, 1.0, 0 )
- What’s the transformation matrix from the world space to the camera reference system?

41. 3D-3D viewing transformation
3D Geometry Pipeline
3D-3D viewing transformation
World space
View space

42. Projection General definition
transform points in n-space to m-space (m<n)
In computer graphics
map 3D coordinates to 2D screen coordinates

43. Projection General definition
transform points in n-space to m-space (m<n)
In computer graphics
map 3D coordinates to 2D screen coordinates
How can we project 3d objects to 2d screen space?

44. How Do We See the World? Let’s design a camera:
idea 1: put a piece of film in front of camera
Do we get a reasonable picture?

45. Pin-hole Camera Add a barrier to block off most of the rays
This reduces blurring
The opening known as the aperture
How does this transform the image?

46. Camera Obscura The first camera Known to Aristotle
Depth of the room is the focal length
Pencil of rays – all rays through a point

47. Perspective Projection
Maps points onto “view plane” along projectors emanating from “center of projection” (COP)

48. Perspective Projection
Maps points onto “view plane” along projectors emanating from “center of projection” (COP)
What’s relationship between 3D points and projected 2D points? 48

49. 3D->2D
Consider the projection of a 3D point on the camera plane

50. 3D->2D
Consider the projection of a 3D point on the camera plane 50

51. 3D->2D Consider the projection of a 3D point on the camera plane
By similar triangles, we can compute how much the x and y coordinates are scaled

52. 3D->2D Consider the projection of a 3D point on the camera plane
By similar triangles, we can compute how much the x and y coordinates are scaled

53. Homogeneous Coordinates
Is this a linear transformation?

54. Homogeneous Coordinates
Is this a linear transformation?
no—division by z is nonlinear

55. Homogeneous Coordinates
Is this a linear transformation?
no—division by z is nonlinear
Trick: add one more coordinate:
homogeneous image
coordinates
homogeneous scene
coordinates

56. Homogeneous Point Revisited
Remember how we said 2D/3D geometric transformations work with the last coordinate always set to one
What happens if the coordinate is not one
We divide all coordinates by w:
If w=1, nothing happens
Sometimes, we call this division step the “perspective divide”

57. The Perspective Matrix
Now we can rewrite the perspective projection equation as matrix-vector multiplications

58. The Perspective Matrix
Now we can rewrite the perspective projection equation as matrix-vector multiplications
This becomes a linear transformation!

59. The Perspective Matrix
Now we can rewrite the perspective projection equation as matrix-vector multiplications
After the division by w, we have

60. Perspective Effects Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)

61. Perspective Effects Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)

62. Perspective Effects Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)

63. Properties of Perspective Proj.
Perspective projection is an example of projective transformation
- lines maps to lines
- parallel lines do not necessary remain parallel
- ratios are not preserved

64. Properties of Perspective Proj.
Perspective projection is an example of projective transformation
- lines maps to lines
- parallel lines do not necessary remain parallel
- ratios are not preserved
One of advantages of perspective projection is that size varies inversely proportional to the distance-looks realistic

65. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?

66. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:

67. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
After perspective transformation, we have

68. Vanishing Points (cont.)
Letting t go to infinity:

69. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:

70. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line

71. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line

72. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line
Same vanishing point!

73. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
Each set of parallel lines intersect at a vanishing point on the PP

74. Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
Each set of parallel lines intersect at a vanishing point on the PP

75. Parallel Projection Center of projection is at infinity
Direction of projection (DOP) same for all points

76. Orthographic Projection
Direction of projection (DOP) perpendicular to view plane

77. Orthographic Projection
Direction of projection (DOP) perpendicular to view plane

78. Properties of Parallel Projection
Not realistic looking
Good for exact measurement
Are actually affine transformation
- parallel lines remain parallel
- ratios are preserved
- angles are often not preserved
Most often used in CAD, architectural drawings, etc. where taking exact measurement is important

79. 3D->2D
Perspective projection from 3D to 2D 79

80. 3D->2D Perspective projection from 3D to 2D
But so far, we have not considered the size of film plane!
We have also not considered visibility problem 80

81. Normalized project space
3D Geometry Pipeline
Object space
World space
View space
z = -1
z = 1
Normalized project space
Image space
Screen space

82. Perspective Projection Volume
The center of projection and the portion of projection plane that map to the final image form an infinite pyramid. The sides of pyramid called clipping planes
Additional clipping planes are inserted to restrict the range of depths

83. Perspective Projection Volume
The center of projection and the portion of projection plane that map to the final image form an infinite pyramid. The sides of pyramid called clipping planes
Additional clipping planes are inserted to restrict the range of depths

84. Perspective Projection Volume

85. Perspective Projection Volume

86. Perspective Projection Volume

87. Perspective Projection Volume

88. OpenGL Perspective-Projection
z = -1
z = 1
Normalized project space
View space
End here
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
gluPerspective(fovy,aspect,dnear, dfar) 88

89. General Perspective-Projection
zfar
(xwmax,ywmax,znear)
(xwmin,ywmin,Znear)
View space
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
Six parameters define six clipping planes!

90. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
near clipping plane

91. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
far clipping plane

92. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
The size of the near plane

93. General Perspective-Projection
B’=(1,1,1) B A
A’=(-1,-1,-1)
H&B equation (10-40)

94. General Perspective-Projection
B’=(1,1,1) B A
A’=(-1,-1,-1)
View space (right-handed)
Normalized project space (left-handed)
A maps to A’, B maps to B’
Keep the directions of x and y axes!

95. General Perspective-Projection
B’=(1,1,1) B A
A’=(-1,-1,-1)
View space (right-handed)
Normalized project space (left-handed)
dnear=-1, dfar=1
xmin=-1,xmax=1,ymin=-1,ymax=1

96. OpenGL Symmetric Perspective-Projection Function
gluPerspective(fovy,aspect,dnear, dfar)
Assume z-axis is the centerline of 3D view frustum!

97. Normalized projection space
3D Geometry Pipeline
Object space
World space
View space
Normalized projection space
Image space

98. Viewport Transformation
Display window
(xvmin,yvmin)
Image space-.>Image space
glViewport(xvmin, yvmin, width, height)

99. Viewport Transformation
- Besides x and y, each pixel has a depth value z, which is stored in depth buffer.
Depth values will be used for visibility testing
The color of the pixel is stored in color buffer
(xvmin,yvmin)
Image space-.>Image space
glViewport(xvmin, yvmin, width, height)

100. Viewport Transformation
From normalized projection coordinates to 3D screen coordinates
dnear= -1, dfar= 1
xmin= -1,xmax= 1
ymin= -1,ymax= 1
dnear= 0, dfar= 1
xmin= xvmin,xmax= xvmax,
ymin= yvmin,ymax= yvmax
glViewport(xvmin, yvmin, width, height)
xmax=xvmin+width,ymax=yvmin+ height

101. Viewport Transformation
0<=z<=1
B’=(1,1,1)
B’’=(xvmax,yvmax,1)
-1<=z<=1
Normalized project space
A’’=(xvmin,yvmin,0)
A’=(-1,-1,-1)
Normalized projection space
3D screen space

102. Viewport Transformation
0<=z<=1
B’=(1,1,1)
B’’=(xvmax,yvmax,1)
-1<=z<=1
Normalized project space
A’’=(xvmin,yvmin,0)
A’=(-1,-1,-1)
Normalized projection space
3D screen space
H&B equation (10-42)

103. Summary: 3D Geometry Pipeline
Object space
World space
View space
Normalized projection space
Image space

104. Normalized project space
Taking Steps Together C M
Object space
World space
View space P V
Normalized project space
Image space

105. OpenGL Codes

106. OpenGL Codes
106

107. OpenGL Codes
107

108. Next Lecture: Hidden Surface Removal