1. Viewing Viewing and viewing space (camera space)
World space to viewing space transformation

2. World to Viewing Coordinates
Graphics display devices are 2D rectangular screens. Hence we need to understand how to transform our 3D world to a 2D surface
Viewing the desired scene is analogous to taking pictures using a camera
We now need to
world
coordinates
viewing
VIEWING TRANSFORMATION 4 4

3. Camera Analogy View is described in terms of:
camera position in world coordinate system
camera direction (viewing direction)
camera orientation: usually defined by the up vector
aperture size: field of view

4. View Coordinate System
specify a view reference point in the world coordinate system. This can be any point along the camera direction, or the camera position itself
specify the view plane normal N - this gives the camera, or Z direction
specify the view-up direction, V - this gives the camera up direction, or Y direction xW yW zW P0 N V xW yW zW P0 N xW yW zW P0 v’ v n
v=v’-(v.n) n 5 5

5. Viewing Coordinate System
We can construct a vector U perpendicular to both V and N, and this will correspond to the Xv axis. How?
We can define U as right, V as up, and N as towards the viewer: a right handed system UV=N
We can also define U as right, V as up and N as into the scene: a left handed system VU=N, in which bigger N values mean points are further away
OpenGL is right handed P0 N V U yW xW zW 8 8

6. View Coordinate System
Some systems (e.g., OpenGL) allow you to specify a ‘look at’ point, Q, from which N is calculated as the direction to the ‘look at’ point from the view reference point xW yW zW P0 Q 6 6

7. World to Viewing Objects must be viewed in the viewing space
This can be done by aligning the view coordinate system with world coordinate system, e.g., view reference point is transformed to world origin, and U, V, N are aligned with X, Y, Z directions through rotations y
(x0, y0, z0) x z 9 9

8. x y z
World to Viewing
Translate view origin to world origin, then align U, V and N axes with X, Y and Z directions by rotation R = Rz. Ry. Rx
rotate around X to bring N into the X-Z plane
rotate around Y to align N with Z
rotate around Z to align V with Y
An easier way to work out the rotation matrix R:
U in world space should be (1,0,0) in view space
V should be (0,1,0)
N should be (0,0,1)
So we have the following equations
1. R*(Ux, Uy, Uz,1)T = (1,0,0,1)
2. R*(Vx, Vy, Vz,1)T = (0,1,0,1)
3. R*(Nx, Ny, Nz,1)T = (0,0,1,1) P0 N V U Ux Uy Uz 1
(Ux, Uy, Uz,1)T = 11 11

9. World to Viewing Transformation
Remember U.U=1, U.V=0, U.N=0, V.N=0, so if we choose the rotation matrix as
The equations 1,2,3 will be satisfied
The rotation matrix is a “change of basis” matrix
So viewing transformation from world space to viewing space is: M = RT
R =

10. What’s Pw in viewing coordinates?x y z u v n
(0,0,1)
Pw=(1,1,0)
Intuitively, P in
viewing coordinate
is (-1,1,1), but how
do we derive it?
1. Translate view origin to world origin with translation vector (0, 0, -1)
2. Multiply Pw by matrix M below to align viewing axes with world axes
M =
So Pw in viewing space is: Pv = M T Pw
u, v, n in world space:
u=(-1,0,0)
v=(0,1,0)
n=(0,0,-1)

11. OpenGL Viewing Coordinate System
The default camera is placed at the coordinate origin of world space (U aligned with the X axis, V aligned with Y, and N aligned with Z), looking along the negative z-direction, and the view plane is perpendicular to the viewing direction xv yv zv 6

12. Projection
We need to transform from a special viewing coordinate system (camera on z-axis pointing along the axis) into a projection coordinate system
viewing
coordinates
PROJECTION TRANSFORMATION
projection
coordinates 3 3

13. Parallel Projection - Two types
In parallel projection, the observer position is at an infinite distance, so the projection lines are parallel
Orthographic parallel projection has view plane perpendicular to direction of projection
Oblique parallel projection has view plane at an oblique angle to direction of projection P1 P2
view plane 14

14. Parallel Projection Calculationxv yv zv
looking along x-axis P
(x,y,z)
(xp,yp,d) yV
viewing space d o
- zV
view plane
yP = y
Similarly, xP = x 15

15. Parallel Projection CalculationSo
x = xp
y = yp
z = d
The projection transformation matrix is simply
If view plane is xoy plane,
Then d=0 xp yp d 1
d/z 0 x y z 1 =
projection space
projection matrix
viewing space 17

16. Perspective Projection
In a perspective projection, object positions are projected onto the view plane along lines which converge at the observer, or Centre of Projection (COP)
Perspective projection gives realistic views, but does not preserve proportions - projections of distant objects are smaller than projections of objects of the same size which are closer to the view plane (fore-shortening) P1 P2
P1’
P2’
view plane
camera 3