1. Viewing Transformation
Tong-Yee Lee

2. Changes of Coordinate System
World coordinate system
Camera (eye) coordinate system

3. Default Camera Position and Orientation
The default camera is with eye at the origin (0,0,0) and the
axis of the pyramid aligned with the z-axis. The eye is looking
down the negative z-axis.

16. In this equation, a is world coordinate, B will convert
a to b (in another coordinate system)

17. Why B is orthonormal?
In above equation, 0 is due to vi.vj=0 i!=j

18. b1=Bw->1aw
aw=Bw->1Tb1=B-w->1b1
b2=Bw->2aw
aw=Bw->2Tb2=B-w->2b2
B-w->1b1=B-w->2b2
Bw->1B-w->1b1= Bw->1B-w->2b2
b1= Bw->1B-w->2 b2= Bw->1B2->w b2

20. Consider a special case as v1=(1,0,0), v2=(0,1,0) and v3(0,0,1)

21. This is same as previous matrix composition by way of
a is coordinate in (v1,v2,v3) system (usually is world coordinate)
b is coordinate in (u1,u2,u3) system
B is easily remembered by carefully checking B
The first row: u1 projects on three axes v1,v2,v3
The second row: u2 projects on three axes v1,v2,v3
The third row: u3 projects on three axes v1,v2,v3
How about B’ for c is coordinate in (w1,w2,w3) for b converted to
c? (i.e. c=B’b)
This is same as previous
matrix composition by way of
transforming to world coordinate

28. X axis vector
Y axis vector
Z axis vector

32. Intuitive Camera Specification

33. Not easy for user to pick up
exact up vector!!
So, we compute v automatically
from up vector.

34. n a b n b’ b

35. n a b
Another way ………………..
a = b’ x n
b = n x a

36. v1=(1,0,0), v2=(0,1,0) and v3(0,0,1)

38. Note that matrix storage order is column major
in OpenGL

39. u (i.e., x) axis
n (i.e.,z) axis
v (i.e., y) axis
Treat yourself (viewer) as a airplane heading to –Zc
Note that: as a viewer is moving, the object is moving in
opposite direction on the viewing plane!!

40. This is z-like rotation

41. n v n’ v’ u n n’ u’
This is x-like rotation
This is y-like rotation

42. How about pitch() and yaw()?
Same stories as roll().
Implementing pitch() and yaw().
void Camera :: pitch (float angle)
{ // pitch the camera through angle degrees around U
float cs = cos( /180 * angle);
float sn = sin( /180 * angle);
Vector3 t(v); // remember old v
v.set(cs*t.x + sn*n.x, cs*t.y + sn*n.y, cs*t.z + sn*n.z);
n.set(-sn*t.x + cs*n.x, -sn*t.y + cs*n.y, -sn*t.z + cs*n.z);
setModelViewMatrix(); }
void Camera :: yaw (float angle)
{ // yaw the camera through angle degrees around V
Vector3 t(n); // remember old v
n.set(cs*t.x + sn*u.x, cs*t.y + sn*u.y, cs*t.z + sn*u.z);
u.set(-sn*t.x + cs*u.x, -sn*t.y + cs*u.y, -sn*t.z + cs*u.z);

43. How about slide()
Sliding a camera means to move it along one of its own axes-that is in the u,v,n direction-without rotating it.
Along n means forward or backward
Along u is left and right
Along v is up and down
Assume slide(delU, delV, delN)

44. Flythrough a Scene!!!