1. GR2 Advanced Computer Graphics AGR
Lecture 4
Viewing Pipeline
Getting Started with OpenGL 1 1

2. The Story So Far ...Lecture 2
We have seen how we can model objects, by transforming them from their local co-ordinate representation into a world co-ordinate system
modelling
co-ordinates
world
MODELLING TRANSFORMATION 2 2

3. The Story So Far...Lecture 3
And we have seen how we can transform from a special viewing co-ordinate system (camera on z-axis pointing along the axis) into a projection co-ordinate system
viewing
co-ordinates
projection
PROJECTION TRANSFORMATION 3 3

4. Completing the Pipeline - Lecture 4
We now need to fill in the missing part
to get
world
co-ordinates
viewing
VIEWING TRANSFORMATION
mod’g
co-ords
world
view’g
proj’n 4 4

5. Viewing Coordinate System - View Reference PointyW
In our world co-ordinate system, we need to specify a view reference point - this will become the origin of the view co-ordinate system
This can be any convenient point, along the camera direction from the camera position
indeed one possibility is the camera position itself P0 xW zW 5 5

6. Viewing Coordinate System - View Plane Normal
Next we need to specify the view plane normal, N - this will give the camera direction, or z-axis direction
Some graphics systems require you to specify N ...
... others (including OpenGL) allow you to specify a ‘look at’ point, Q, from which N is calculated as direction to the ‘look at’ point from the view reference point yW P0 N xW zW yW P0 Q xW zW 6 6

7. Viewing Coordinate System - View Up Direction
Finally we need to specify the view-up direction, V - this will give the y-axis direction V yW P0 N xW zW 7 7

8. Viewing Co-ordinate System
This gives us a view reference point P0,and vectors N (corresponding to zV) and V (corresponding to yV)
We can construct a vector U perpendicular to both V and N, and this will correspond to the xV axis
How? V U yW P0 N xW zW 8 8

9. Transformation from World to Viewing Co-ordinates
Given an object with positions defined in world co-ordinates, we need to calculate the transformation to viewing co-ordinates
The view reference point must be transformed to the origin, and lines along the U, V, N directions must be transformed to lie along the x, y, z directions 9 9

10. Transformation from World to Viewing Co-ordinates
Translate so that P0 lies at the origin V yW U N P0
(x0, y0, z0) xW zW
- apply translation by (-x0, -y0, -z0)
T = x0 y0 z0 10 10

11. Transformation from World to Viewing Co-ordinates
Apply rotations so that the U, V and N axes are aligned with the xW, yW and zW directions
This involves three rotations Rx, then Ry, then Rz
first rotate around xW to bring N into the xW-zW plane
second, rotate around yW to align N with zW
third, rotate around zW to align V with yW
Composite rotation R = Rz. Ry. Rx 11 11

12. Rotation Matrix
Fortunately there is an easy way to calculate R, from U, V and N:
R =
u1 u2 u3 0
v1 v2 v3 0
n1 n2 n3 0
where U = (u1 u2 u3 )T
etc 12 12

13. Viewing Transformation
Thus the viewing transformation is:
M = R . T
This transforms object positions in world co-ordinates to positions in the viewing co-ordinate system..
.. with camera pointing along negative z-axis at a view plane parallel to x-y plane
We can then apply the projection transformation 13 13

14. Viewing Pipeline So Far
We now should understand this viewing pipeline
mod’g
co-ords
world
view’g
proj’n 14 14

15. Clipping
Next we need to understand how the clipping to the view volume is performed
Recall that with perspective projection we defined a view frustum outside of which we wanted to clip points and lines, etc
The next slide is from lecture 3 ... 15 15

16. View Frustum - Perspective Projection
back
plane
view frustum
view window
camera
front
plane zV 16 16

17. Clipping to View Frustum
It is quite easy to clip lines to the front and back planes (just clip in z)..
.. but it is difficult to clip to the sides because they are ‘sloping’ planes
Instead we carry out the projection first which converts the frustum to a rectangular parallelepiped (ie a cuboid) 17 17

18. Clipping for Parallel Projection
In the parallel projection case, the viewing volume is already a rectangular parallelepiped
view window
back
plane
front zV
view volume 18 18

19. Normalized Projection Co-ordinates
Final step before clipping is to normalize the co-ordinates of the rectangular parallelepiped to some standard shape
for example, in some systems, it is the cube with limits +1 and -1 in each direction
This is just a scale transformation
Clipping is then carried out against this standard shape 19 19

20. Viewing Pipeline So Far
Our pipeline now looks like:
mod’g
co-ords
world
view’g
proj’n
normalized
projection
co-ordinates
NORMALIZATION TRANSFORMATION 20 20

21. And finally...
The last step is to position the picture on the display surface
This is done by a viewport transformation where the normalized projection co-ordinates are transformed to display co-ordinates, ie pixels on the screen 21 21

22. Viewing Pipeline - The End
A final viewing pipeline is therefore:
mod’g
co-ords
world
view’g
proj’n
normalized
projection
co-ordinates
device
co-ordinates
DEVICE TRANSFORMATION 22 22

23. Interlude
Why does a mirror reflect left-right and not up-down?

24. OpenGL - Getting Started

25. What is OpenGL?
OpenGL provides a set of routines (API) for advanced 3D graphics
derived from Silicon Graphics GL
acknowledged industry standard, even on PCs (OpenGL graphics cards available)
integrates 3D drawing into X (and other window systems such as Windows NT)
draws simple primitives (points, lines, polygons) but NOT complex primitives such as spheres
provides control over transformations, lighting, etc
Mesa is publically available equivalent 4 4

26. Geometric Primitives
Defined by a group of vertices - for example to draw a triangle:
glBegin (GL_POLYGON);
glVertex3i (0, 0, 0);
glVertex3i (0, 1, 0);
glVertex3i (1, 0, 1);
glEnd();
See Chapter 2 of the OpenGL Programming Guide 5 5

27. Viewing OpenGL maintains two matrix transformation modes
MODELVIEW
to specify modelling transformations, and transformations to align camera
PROJECTION
to specify the type of projection (parallel or perspective) and clipping planes
See Chapter 3 of OpenGL Programming Guide 6 6

28. OpenGL Utility Library (GLU)
Useful set of higher level utility routines to make some tasks easier
written in terms of OpenGL and provided with the OpenGL implementation
for example, gluLookAt() is a way of specifying the viewing transformation
Described within the OpenGL Programming Guide
eg gluLookAt() is described in Chap 3, pp19-21 7 7

29. OpenGL Utility Toolkit (GLUT)
Set of routines to provide an interface to the underlying windowing system - plus many useful high-level primitives (even a teapot - glutSolidTeapot()!)
See Appendix D of OpenGL Guide
Allows you to write ‘event driven’ applications
you specify call back functions which are executed when an event (eg window resize) occurs 8 8

30. How to Get Started Look at the GR2/AGR resources page: Points you to:
resources.html
Points you to:
example programs
information about GLUT
information about OpenGL
a simple exercise