1. CSCE 441 Computer Graphics 3-D Viewing
Dr. Jinxiang Chai

2. Outline 3D Viewing Required readings: HB 7-1 to 7-10
Compile and run the codes in page 388 1

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

4. Taking Pictures Using A Real Camera
Steps:
- Identify interesting objects
- Rotate and translate the camera to a desire 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

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
Image space 4

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

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

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

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

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

16. 3D Geometry Pipeline World space View space
Now look at how we would compute the world->eye transformation
World space
View space

17. 3D Geometry Pipeline World space View space
Now look at how we would compute the world->eye transformation
Rotate&translate
World space
View space

18. Camera Coordinate

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

20. 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

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

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

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

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

25. Review: 3D Coordinate Trans.
Transform object description from to p 24

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

27. 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

28. Viewing Transformation
Trick: find T-1 taking object from camera to world
H&B equation (7-4)

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

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

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

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

33. Viewing Trans: gluLookAt
Mapping from world to eye coordinates
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Stop here.
H&B equation (7-1)

34. 3D-3D viewing transformation
3D Geometry Pipeline
3D-3D viewing transformation
World space
View space

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

36. 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?

37. 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?

38. 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?

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

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

41. 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? 40

42. 3D->2D
Consider the projection of a 3D point on the camera plane

43. 3D->2D
Consider the projection of a 3D point on the camera plane 42

44. 3D->2D Consider the projection of a point on the camera plane
By similar triangles, we can compute how much the x and y coordinates are scaled

45. 3D->2D Consider the projection of a point on the camera plane
By similar triangles, we can compute how much the x and y coordinates are scaled

46. Homogeneous Coordinates
Is this a linear transformation?

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

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

49. 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”

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

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

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

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

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

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

56. 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

57. 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

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

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

60. 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

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

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

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

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

65. 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!

66. 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

67. 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
How many vanishing points are there?

68. 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
How many vanishing points are there?

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

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

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

72. Oblique Projection
DOP not perpendicular to view plane A’ A O

73. Oblique Projection
DOP not perpendicular to view plane A’ A O

74. 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

75. 3D->2D
Perspective projection from 3D to 2D 74

76. 3D->2D Perspective projection from 3D to 2D
But so far, we have not considered the size of film plane! 75

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

78. 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

79. OpenGL Perspective-Projection
Normalized project space
View space
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
gluPerspective(fovy,aspect,dnear, dfar) 78

80. 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!

81. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
Left-vertical clipping plane

82. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
right-vertical clipping plane

83. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
bottom-horizontal clipping plane

84. General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
top-horizontal clipping plane

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

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

87. 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!

88. General Perspective-Projection
B’=(1,1,1) B A
A’=(-1,-1,-1)
H&B equation (7-40)

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

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

91. Viewport Transformation
(xmin,ymin)
Image space-.>Image space
gluViewport(xmin, ymin, width, height)

92. Viewport Transformation
(xmin,ymin)
Besides x and y, each pixel has a depth value z!
Image space-.>Image space
gluViewport(xmin, ymin, width, height)

93. Viewport Transformation
B’=(1,1,1)
B’’=(xvmax,yvmax,1)
Normalized project space
A’’=(xvmin,yvmin,0)
A’=(-1,-1,-1)
Normalized project space
Image space

94. Viewport Transformation
B’=(1,1,1)
B’’=(xvmax,yvmax,1)
Normalized project space
A’’=(xvmin,yvmin,0)
A’=(-1,-1,-1)
Normalized project space
Image space
H&B equation (7-42)

95. Summary: 3D Geometry Pipeline
Object space
World space
View space
Normalized project space
Image space

96. Normalized project space
Taking Steps Together
Object space
World space
View space
Normalized project space
Image space

97. OpenGL Codes