681 Introduction to Computer Graphics

1
Orientation
Rotation about principle axes - fixed angles
Rotation about objects axes - Euler angles
Axis-angle rotation
Quaternion
2
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
3
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)
4
Gimbal lock
From some orientations, cant do some rotations
(0,90,0)
Cant rotate around x-axis
5
Euler angles
Rotate about axes of object
Can use most any triple of axes
Roll, Pitch, Yaw
(10, 90, -45)
6
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)
7
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)
8
Quaternions
Keep axis-angle orientation as 4-tuple
(sin(t/2), cos(t/2)(x,y,z))