Technical Background

1
Technical Background
Display pipeline
Transformations and round-off errors
Orientation representation
Quaternions
2
Viewing Pipeline
Transformation between spaces
Object space
transformations
transformations
World space
Eye space
Map to eye space
perspective
Clipping space
Perspective divide
Image space
Viewport mapping
Screen space
3
Ray Tracing
Object space
transformations
transformations
World space
Eye space
Map to eye space
Trace rays
Screen space
4
Transformations
Px
b
d
a
c
Py
i
f
h
e
Pz
m
l
j
k
1
q
p
o
n
5
Transformations
0
Tx
1
0
0
0
Sx
0
Ty
1
0
0
0
0
0
Sy
Tz
1
0
0
0
Sz
0
0
1
0
0
0
1
0
0
0
6
Rotation
1 0 0 0 cos(q) -sin(q) 0
sin(q) cos(q)
We know how to rotate about the global axes
cos(q) 0 sin(q) 0 1
0 -sin(q) 0 cos(q)
cos(q) -sin(q) 0 sin(q) cos(q) 0 0
0 1
7
Transformation round-off errors
Rotate 5 degrees every frame
M 5 degree Rotation matrix
Apply M to data in world space
8
Transformation round-off errors
Rotate 5 degrees every frame
m 5 degree
c 0 degrees
c c m
M rotation matrix Of c degrees
Apply M to data in world space
9
Transformation round-off errors
Rotate 5 degrees every frame
M 5 degree Rotation matrix
C Identity matrix
C C M
Apply C to data
10
Transformation round-off errors
Rotate 5 degrees every frame
m 5 degree
M 5 degree Rotation matrix
M 5 degree Rotation matrix
c 0 degrees
c c m
C Identity matrix
Apply M to data in world space
M rotation matrix Of c degrees
C C M
Apply M to data in world space
Apply C to data
11
Orientation
Rotation about principle axes - fixed angles
Rotation about objects axes - Euler angles
Axis-angle rotation
Quaternion
12
Orientation
1 0 0 0 cos(q) -sin(q) 0
sin(q) cos(q)
We know how to rotate about the global axes
cos(q) 0 sin(q) 0 1
0 -sin(q) 0 cos(q)
cos(q) -sin(q) 0 sin(q) cos(q) 0 0
0 1
13
Fixed angles
Rotate about global axes in a fixed order
Rotating about global axes is what the rotation
matrices do
Can use most any triple of axes
Rotate about x, then y, then z
(10, 90, -45)
14
Gimbal lock
From some orientations, cant do some rotations
(0,90,0)
Cant rotate around x-axis
15
Euler angles
Rotate about axes of object
Can use most any triple of axes
Roll, Pitch, Yaw
(10, 90, -45)
16
Equivalence of Fixed angles and Euler angles
Ru(a) Rx (a)
Rv (b)Ru (a) Rx(a)Ry (b)Rx (- a)Rx (a) Rx
(a)Ry (b)
Rw (g) Rv (b)Ru (a) Rx (a)Ry (b) Rz (g)
17
Quaternions
Keep axis-angle orientation as 4-tuple
Q (s, x, y, z) (s,v)
Q1Q2 (s1,v1)(s2,v2) (s1s2v1v2 , s1v2
s2v1 v1xv2)
Q1Q2 (s1,v1)(s2,v2) (s1s2, v1v2)
18
Quaternions
Keep axis-angle orientation as 4-tuple
(sin(t/2), cos(t/2)(x,y,z))