1. 3D Viewing
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2. 3d Rendering Pipeline
3D Primitives
Model Transformation
Lighting
Viewing Transformation
This is a pipelined sequence of operations to draw a 3D primitive into a 2D image for direct illumination
Projection Transformation
Clipping
Viewport Transformation
Scan Conversion
Image
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3. In Pipeline Transform into3d world coordinate system 3D Primitives
Model Transformation
Transform into3d world coordinate system
Lighting
Viewing Transformation
Projection Transformation
Clipping
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

4. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Viewing Transformation
Projection Transformation
Clipping
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

5. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Projection Transformation
Clipping
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

6. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Transform into 2D viewing coordinate system
Projection Transformation
Clipping
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

7. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Transform into 2D viewing coordinate system
Projection Transformation
Clip primitives outside window’s view
Clipping
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

8. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Transform into 2D viewing coordinate system
Projection Transformation
Clip primitives outside window’s view
Clipping
Transform into viewport
Viewport Transformation
Scan Conversion
Image
cgvr.korea.ac.kr

9. In Pipeline Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Transform into 2D viewing coordinate system
Projection Transformation
Clip primitives outside window’s view
Clipping
Transform into viewport
Viewport Transformation
Draw pixels(includes texturing, hidden surface etc.)
Scan Conversion
Image
cgvr.korea.ac.kr

10. Transformation Transform into3d world coordinate system
3D Primitives
Model Transformation
Transform into3d world coordinate system
Illustrate according to lighting and reflectance
Lighting
Transform into 3D viewing coordinate system
Viewing Transformation
Transform into 2D viewing coordinate system
Projection Transformation
Clip primitives outside window’s view
Clipping
Transform into viewport
Viewport Transformation
Draw pixels(includes texturing, hidden surface etc.)
Scan Conversion
Image
cgvr.korea.ac.kr

11. Transformation 3D Object Coordinate 3D World Coordinate
P(x, y, z)
3D Object Coordinate
3D Viewing Coordinate
Model Transformation
3D World Coordinate
Viewing Transformation
3D Viewing Coordinate
Projection Transformation
3D Object Coordinate
2D Projection Coordinate
Viewport Transformation
3D World Coordinate
2D Device Coordinate
p(x’, y’)
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12. Viewing Transformation
P(x, y, z)
3D Object Coordinate
Model Transformation
3D World Coordinate
Viewing
Transformation
Viewing Transformation
3D Viewing Coordinate
Projection Transformation
2D Projection Coordinate
Viewport Transformation
2D Device Coordinate
p(x’, y’)
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13. Viewing Transformation
Mapping from world to Viewing coordinates
Origin moves to eye position
Up vector maps to Y axis
Right vector maps to X axis Y Z
Camera X
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14. Transformation from WC to VC
Transformation sequences
1. Translate the view reference point to the origin of the WC system
2. Apply rotations to align the xv, yv, and zv axes with the world axes
General sequence of translate-rotate transformation
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15. Transformation from WC to VC (cont’)
Translation
view reference point(x0, y0, z0)
Rotation
rotate around the world xw axis to bring zv into the xwzw plane
rotate around the world yw axis to align the zw and zv axis
final rotation is about the zw axis to align the yw and yv axis
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16. Transformation from WC to VC (cont’)
Rotation by uvn system
Calculate unit uvn vectors
N : view-plane normal vector
V : view-up vector
U : perpendicular to both N and V
Form the composite rotation matrix
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17. Camera Models The most common model is pin-hole camera
All captured light rays arrive along paths toward focal point without lens distortion (everything is in focus)
Sensor response proportional to radiance
Other models consider…
Depth of field
Motion blur
Lens distortion
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18. Viewing Parameters Position Orientation Aperture Film plane
Eye position(px, py, pz)
Orientation
View direction(dx, dy, dz)
Up direction(ux, uy, uz)
Aperture
Field of view(xfov, yfov)
Film plane
“look at” point
View plane normal
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19. Viewing Coordinate Canonical coordinate system
Convention is right-handed (looking down – z axis)
Convention for projection, clipping, etc.
Viewing up vector maps to Y axis Y
Viewing back vector maps to Z axis (potting out of page)
Viewing right vector maps to X axis X
cgvr.korea.ac.kr

20. Viewing Transformation
Transformation matrix maps camera basis vectors to canonical vectors in viewing coordinate system
(0, 1, 0)
Back Up
Matrix
(1, 0, 0)
Eye
Right
(0, 0, 1)
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21. Viewing Transformation
P(x, y, z)
3D Object Coordinate
Model Transformation
3D World Coordinate
Viewing Transformation
3D Viewing Coordinate
Projection
Transformation
Projection Transformation
2D Projection Coordinate
Viewport Transformation
2D Device Coordinate
p(x’, y’)
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22. Projection General definition In computer graphics
Transform points in n-space to m-space(m<n)
In computer graphics
Map viewing coordinates to 2D screen coordinates
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23. Taxonomy of Projections
Planar geometric projection
Parallel
Perspective
Orthographic
Oblique
Top
Front
Side
Axonometric
Cabinet
Cavalier
Other
One-point
Two-point
Three-point
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24. Parallel & Perspective
Parallel Projection
Perspective Projection
cgvr.korea.ac.kr

25. Taxonomy of Projections
Planar geometric projection
Parallel
Perspective
Orthographic
Oblique
Top
Front
Side
Axonometric
Cabinet
Cavalier
Other
One-point
Two-point
Three-point
cgvr.korea.ac.kr

26. Parallel Projection Center of projection is at infinity
Direction of projection (DOP) same for all points
DOP
View Plane
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27. Taxonomy of Projections
Planar geometric projection
Parallel
Perspective
Orthographic
Oblique
Top
Front
Side
Axonometric
Cabinet
Cavalier
Other
One-point
Two-point
Three-point
cgvr.korea.ac.kr

28. Parallel Projection View Volume
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29. Orthographic & Oblique
Orthographic parallel projection
the projection is perpendicular to the view plane
Oblique parallel projection
The projectors are inclined with respect to the view plane
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30. Orthographic Projections
DOP perpendicular to view plane
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31. Orthographic Projections
DOP perpendicular to view plane
Front
Top
Side
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32. Orthographic Coordinates
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33. Oblique Projections DOP not perpendicular to view plane Cavalier
(DOP at 45 )
Cabinet
(DOP at 63.4 )
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34. Oblique Projections DOP not perpendicular to view plane
Cavalier projection
Cabinet projection
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35. Parallel Projection Matrix
General parallel projection transformation
Where L1 is the inverse of tan α , which is also the value of L when z=1
cgvr.korea.ac.kr

36. Parallel Projection Matrix
General parallel projection transformation
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37. Parallel Projection Matrix
cgvr.korea.ac.kr

38. Taxonomy of Projections
Planar geometric projection
Parallel
Perspective
Orthographic
Oblique
Top
Front
Side
Axonometric
Cabinet
Cavalier
Other
One-point
Two-point
Three-point
cgvr.korea.ac.kr

39. Perspective Projection
Map points onto “view plane” along “projectors” emanating from “center of projection”(cop)
Projectors
Center of Projection
View Plane
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40. Perspective Projection
How many vanishing point?
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41. Perspective Projection
How many vanishing point?
Three-point perspective
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42. Perspective Projection
How many vanishing point?
Three-point perspective
Two-point perspective
cgvr.korea.ac.kr

43. Perspective Projection
How many vanishing point?
Three-point perspective
Two-point perspective
One-point perspective
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44. Perspective Projection View Volume
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45. Perspective Projection
Compute 2D coordinates from 3D coordinates with similar triangles
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46. Perspective Projection
Compute 2D coordinates from 3D coordinates with similar triangles
cgvr.korea.ac.kr

47. Perspective Projection Matrix
4x4 matrix representation?
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48. Perspective Projection Matrix
4x4 matrix representation?
cgvr.korea.ac.kr

49. Perspective Projection Matrix
Orthographic
projection
Perspective
transformation
Center of Projection on the x axis
Center of Projection on the y axis
cgvr.korea.ac.kr

50. Perspective Projection Matrix
2-point perspectives
3-point perspectives
cgvr.korea.ac.kr

51. Taxonomy of Projections
Planar geometric projection
Parallel
Perspective
Orthographic
Oblique
Top
Front
Side
Axonometric
Cabinet
Cavalier
Other
One-point
Two-point
Three-point
cgvr.korea.ac.kr

52. Perspective vs. Parallel
Perspective projection
+ Size varies inversely with distance – looks realistic
– Distance and angles are not(in general) preserved
– Parallel line do not (in general) remain parallel
Parallel projection
+ Good for exact measurements
+ Parallel lines remain parallel
– Angles are not (in general) preserved
– Less realistic looking
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53. Classical Viewing
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54. Thanks…
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