1. Advanced Computer Graphics Three Dimensional Viewing

2. Overview
Viewing a 3D scene
Projections
Parallel and perspective

3. Overview
Surface rendering

4. Overview
Exploded and cutaway views

5. Overview
3D and stereoscopic viewing

6. 3D Viewing Pipeline MC DC Modeling Transformation
Viewport Transformation NC WC
Normalization
Transformation
and Clipping
Viewing Transformation VC PC
Projectionn Transformation

7. Viewing Coordinates
Generating a view of an object in 3D is similar to photographing the object.
Whatever appears in the viewfinder is projected onto the flat film surface.
Depending on the position, orientation and aperture size of the camera corresponding views of the scene is obtained.

8. Specifying The View Coordinatesxw zw yw xv zv yv
P0=(x0 , y0 , z0)
For a particular view of a scene first we establish viewing-coordinate system.
A view-plane (or projection plane) is set up perpendicular to the viewing z-axis.
World coordinates are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.

9. Specifying The View Coordinates
To establish the viewing reference frame, we first pick a world coordinate position called the view reference point.
This point is the origin of our viewing coordinate system. If we choose a point on an object we can think of this point as the position where we aim a camera to take a picture of the object.

10. Specifying The View Coordinates
Next, we select the positive direction for the viewing z-axis, and the orientation of the view plane, by specifying the view-plane normal vector, N.
We choose a world coordinate position P and this point establishes the direction for N.
OpenGL establishes the direction for N using the point P as a look at point relative to the viewing coordinate origin. yv xv xv yw zv N P0 P xw zw

11. Specifying The View Coordinates
Finally, we choose the up direction for the view by specifying view-up vector V.
This vector is used to establish the positive direction for the yv axis.
The vector V is perpendicular to N.
Using N and V, we can compute a third vector U, perpendicular to both N and V, to define the direction for the xv axis. yv xv V yw zv N P0 P xw zw

12. Specifying The View Coordinates
To obtain a series of views of a scene , we can keep the the view reference point fixed and change the direcion of N. This corresponds to generating views as we move around the viewing coordinate origin. P0 N N

13. Transformation From World To Viewing Coordinates
Conversion of object descriptions from world to viewing coordinates is equivalent to transformation that superimpoes the viewing reference frame onto the world frame using the translation and rotation. xv yv zv yw xw zw

14. Transformation From World To Viewing Coordinates
First, we translate the view reference point to the origin of the world coordinate system xv yv zv yw xw zw

15. Transformation From World To Viewing Coordinates
Second, we apply rotations to align the xv,, yv and zv axes with the world xw, yw and zw axes, respectively. yw yv zv xv yv zv xv xw zw

16. Transformation From World To Viewing Coordinates
If the view reference point is specified at word position (x0, y0, z0), this point is translated to the world origin with the translation matrix T.

17. Transformation From World To Viewing Coordinates
The rotation sequence requires 3 coordinate-axis transformation depending on the direction of N.
First we rotate around xw-axis to bring zv into the xw -zw plane.

18. Transformation From World To Viewing Coordinates
Then, we rotate around the world yw axis to align the zw and zv axes.

19. Transformation From World To Viewing Coordinates
The final rotation is about the world zw axis to align the yw and yv axes.

20. Transformation From World To Viewing Coordinates
The complete transformation from world to viewing coordinate transformation matrix is obtaine as the matrix product

21. Transformation From World To Viewing Coordinates: An Example For 2d System
Θ=300
P0=(4,3) x

22. Transformation From World To Viewing Coordinates: An Example For 2d System
Translation: y P 2 x′ 2 y′
Θ=300 P0 x

23. Transformation From World To Viewing Coordinates: An Example For 2d System
Rotation y x x′ y′ P P0

24. Transformation From World To Viewing Coordinates: An Example For 2d System
New coordinates

25. Transformation From World To Viewing Coordinates: An Example For 2d System
Alternative Method y P 1 x′ y′ 1 n v
Θ=300 P0 x

26. Projections
Once WC description of the objects in a scene are converted to VC we can project the 3D objects onto 2D view-plane.
Two types of projections:
-Parallel Projection
-Perspective Projection

27. Classical Viewings
Hand drawings : Determined by a specific relationship between the object and the viewer.

28. Parallel Projections
Coordinate Positions are transformed to the view plane along parallel lines.
View
Plane P2
P′2 P1
P′1

29. Parallel Projections
Orthographic parallel projection The projection is perpendicular to the view plane.
Oblique parallel projecion The parallel projection is not perpendicular to the view plane.

30. Orthographic Parallel Projection

31. Oblique Parallel Projection
The projectors are still ortogonal to the projection plane
But the projection plane can have any orientation with respect to the object.
It is used extensively in architectural and mechanical design.

32. Oblique Parallel Projection
Preserve parallel lines but not angles
Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object.
Dimetric view : Symmetric with two faces.
Trimetric view : General case.

33. Oblique Parallel Projection
Preserve parallel lines but not angles
Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object.
Dimetric view : Symmetric with two faces.
Trimetric view : General case.

34. Perspective Projections
First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance
Objects closer to viewer look larger
Parallel lines appear to converge to single point

35. Perspective Projections
In perspective projection object positions are transformed to the view plane along lines that converge to a point called the projection reference point (or center of projection)

36. Perspective Projections
In the real world, objects exhibit perspective foreshortening: distant objects appear smaller
The basic situation:

37. Perspective Projections
When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world:
How tall should this bunny be?

38. Perspective Projections
The geometry of the situation is that of similar triangles. View from above:
P (x, y, z) X Z
(0,0,0)
x′ = ?
View plane
(xp, yp) d

39. Perspective Projections
Desired result for a point [x, y, z, 1]T projected onto the view plane:

40. View Volumes
When a camera used to take a picture, the type of lens used determines how much of the scene is caught on the film.
In 3D viewing, a rectangular view window in the view plane is used to the same effect. Edges of the view window are parallel to the xv-yv axes and window boundary positions are specified in viewing coordinates.

41. View Volumes Parallel Projection Parallel Projection
View volume (frustum)
window zv
Back Plane
window
Front Plane
Back Plane
Projection Reference Point
Front Plane
Parallel Projection
Parallel Projection
Perspective Projection

42. Clipping
An algorithm for 3D clipping identifies and saves all surface segments within the view volume for display.
All parts of object that are outside the view volume are discarded.

43. Clipping Lines
Just like the case in two dimensions, clipping removes objects that will not be visible from the scene
The point of this is to remove computational effort
3-D clipping is achieved in two basic steps
Discard objects that can’t be viewed
i.e. objects that are behind the camera, outside the field of view, or too far away
Clip objects that intersect with any clipping plane

44. Discard Objects
Discarding objects that cannot possibly be seen involves comparing an objects bounding box/sphere against the dimensions of the view volume
Can be done before or after projection

45. Clipping Polygon Surface
To clip a polygon surface, we can clip the individual polygon edges.
First we test the coordinate extends against each boundary of the view volume to determine whether the object is completely inside or completely outside of that boundary.
If the object has intersection with the boundary then we apply intersection calculations.

46. Clipping Polygon Surface
The projection operation can take place before the view- volume clipping or after clipping.
All objects within the view volume map to the interior of the specified projection window.
The last step is to transform the window contents to a 2D view port.

47. Normalisation
The transformed volume is then normalised around position (0, 0, 0) and the z axis is reversed

48. Dividing Up The World
Similar to the case in two dimensions, we divide the world into regions
This time we use a 6-bit region code to give us 27 different region codes
The bits in these regions codes are as follows:
bit 6
Far
bit 5
Near
bit 4
Top
bit 3
Bottom
bit 2
Right
bit 1
Left

49. Region Codes

50. 3D Line Clipping Example (cont…)
When then simply continue as per the two dimensional line clipping algorithm

51. Clipping Polygon Surface
Viev volume

52. 3D Polygon Clipping (cont…)
In this case we first try to eliminate the entire object using its bounding volume
Next we perform clipping on the individual polygons using the Sutherland-Hodgman algorithm we studied previously

53. OpenGL Projection Commands

54. OpenGL Look-At Function
OpenGL utility function
VRP: eyePoint (eyex, eyey, eyez)
VPN: – ( atPoint – eyePoint ) (atx, aty, atz) – (eyex, eyey, eyez)
VUP: upPoint – eyePoint (upx, upy, upz)
gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz);
look-at positioning

55. Projections in OpenGL Angle of view, field of view :
Only objects that fit within the angle of view of the camera appear in the image
View volume, view frustum :
Be clipped out of scene
Frustum – truncated pyramid