1. UBI 516 Advanced Computer Graphics Three Dimensional Viewing
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2. Overview
Viewing a 3D scene
Projections
Parallel and perspective

3. Overview Depth cueing and hidden surfaces
Identifying visible lines and surfaces

4. Overview
Surface rendering

5. Overview
Exploded and cutaway views

6. Overview
3D and stereoscopic viewing

7. 3D Viewing Pipeline MC DC Modeling Transformation
Viewport Transformation NC WC
Normalization
Transformation
and Clipping
Viewing Transformation VC PC
Projectionn Transformation

8. 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.

9. 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.

10. 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.

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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.

18. 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.

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

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

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

22. Transformation From World To Viewing Coordinates
Another method for generating the rotation-transformation matrix is to calculate uvn vectors and obtain the composite rotation matrix directly. Given the vectors and , these unit vectors are calculated as

23. Transformation From World To Viewing Coordinates
This method also automatically adjusts the direction for so that is perpendicular to The rotation matrix for the viewing transformation is then

24. Transformation From World To Viewing Coordinates
The matrix for translating the viewing origin to the world origin is

25. Transformation From World To Viewing Coordinates
The composite matrix for the viewing transformation is then

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

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

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

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

30. 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

31. 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

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

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

34. 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.

35. Orthographic Parallel Projection
The orthographic transformation

36. Orthographic Parallel Projection

37. 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.

38. 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.

39. 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.

40. Oblique Parallel Projectionyv
(xp, yp) α
(x, y, z) L φ xv
(x, y) zv

41. Oblique Parallel Projection
The oblique transformation

42. Oblique Parallel Projection

43. 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

44. 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)

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

46. 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?

47. 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

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

49. Perspective Projections

50. Perspective Projections

51. Projection Matrix
We talked about geometric transforms, focusing on modeling transforms
Ex: translation, rotation, scale, gluLookAt()
These are encapsulated in the OpenGL modelview matrix
Can also express projection as a matrix
These are encapsulated in the OpenGL projection matrix

52. 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.

53. 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

54. 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.

55. Clipping Lines
To clip a line against the view volume, we need to test the relative position of the line using the view volume’s boundary plane equation.
An end point (x,y,z) of a line segment is outside a boundary plane if
where A, B, C and D are the plane parameters for that boundary.

56. 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.

57. 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.

58. Clipping Polygon Surface
Viev volume

59. Steps For Normalized View Volumes
A scene is constructed by transforming object descriptions from modeling coordinates to wc.
The world descriptions are converted to viewing coordinates.
The viewing coordinates are transformed to projection coordinates which effectively converts the view volume into a rectangular parallelepiped.
The parallelepiped is mapped into the unit cube called normalized projection coordinate system.
A 3D viewport within the unit cube is constructed.
Normalized projection coordinates are converted to device coordinates for display.

60. Normalized View Volumesy x
(Xwmax, ywmax, zback)
(Xvmax, yvmax, zvmax) z
(Xwmin, ywmin, zfront)
Parallelepiped
View Volume
(Xvmin, yvmin, zwmin)
Unit Cube

61. Orthogonal Projection Normalization

62. Oblique Projection Normalization
Angles of projection
 for x axis
 for y axis
Shearing matrix H(, )  

63. Oblique Projection Normalization
Finished ? No, this is a sheared view volume, so we have to apply orthogonal transformation :
P=Porth STH

64. Perspective Projection Normalization
Perspective Normalization is Trickier

65. Perspective Projection Normalization
Consider N =
After multiplying:
p’ = Np

66. Perspective Projection Normalization
After dividing by w’, p’ -> p’’

67. Perspective Projection Normalization
Quick Check
If x = z
x’’ = -1
If x = -z
x’’ = 1

68. Perspective Projection Normalization
What about z?
if z = zmax
if z = zmin
Solve for a and b such that zmin -> -1 and zmax ->1
Resulting z’’ is nonlinear, but preserves ordering of points
If z1 < z2 … z’’1 < z’’2

69. Perspective Projection Normalization
We did it. Using matrix, N
Perspective viewing frustum transformed to cube
Orthographic rendering of cube produces same image as perspective rendering of original frustum

70. OpenGL Projection Commands

71. 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

72. 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

73. Projections in OpenGL

74. Perspective in OpenGL Specification of a frustum
near, far: positive number !!
 zmax = – far
 zmin = – near
glMatrixMode(GL_PROJECTION);
glLoadIdentity( );
glFrustum(xmin, xmax, ymin, ymax, near, far);

75. Perspective in OpenGL Specification using the field of view
fov: angle between top and
bottom planes
fovy: the angle of view in the
up (y) direction
aspect ratio: width / height
glMatrixMode(GL_PROJECTION);
glLoadIdentity( );
gluPerspective(fovy, aspect, near, far);

76. Parallel Viewing in OpenGL
Orthographic viewing function
OpenGL provides only this parallel-viewing function
near < far !!
 no restriction on the sign
 zmax = – far
 zmin = – near
glMatrixMode(GL_PROJECTION);
glLoadIdentity( );
glOrtho(xmin, xmax, ymin, ymax, near, far);

77. Optional Clipping Planes
glClipPlane(id, PlaneParameters); glEnable(id); // id = GL_CLIP_PLANE0, GL_CLIP_PLANE1, ... // PlaneParameters = A,B,C and D of the plane glDisable(id);