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ORIENTATION
Interpolating rotations is difficult
Use Quaternions
Rick Parent - CIS681
Object Representation
Define object in world space
Object space data
Scale
Rotation
Translation
Desired operations
Interpolation between transformations
Concatenation of one transformation after another
Handle scale, rotation, translation, independently
Rotation deserves special attention!
Rick Parent - CIS681
Repeated Rotations: Error Management
Task: Rotate an object some Dq every frame
Issue: Avoiding accumulated roundoff error
Rick Parent - CIS681
Repeated Rotations: Error Management
Method 1
M = create_rotation_matrix(Dq)
Object = apply M to Object <= repeat
Method 2
D = create_rotation_matrix(Dq)
M create_rotation_matrix(q)
M=DM
<= repeat
Object = apply M to object
Method 3
q = q + Dq
M = create_rotation_matrix(q) <= repeat
Object = apply M to object
Rick Parent - CIS681
Orientation Representation
orientation
Rick Parent - CIS681
Interpolation
O1
O 1.5
O2
Rick Parent - CIS681
Concatenation
O1
O2
Rick Parent - CIS681
Orientation Representation
Rotation Matrix
Fixed Angles
Euler Angles
Axis-Angle
Quaternion
Rick Parent - CIS681
Rotation Matrices
a b c 0
d e f 0
g h i 0
0 0 0 1
9 values but 3 degrees of freedom
Euler’s rotation theorem: An arbitrary rotation
may be described by only three parameters.
Rick Parent - CIS681
Rotation Matrices
Can’t interpolate
rotation matrices
0 -1 0 0 0 1 0 0
1 0 0 0 -1 0 0 0
0 0 1 0 0 0 1 0
0 0 0 1
0 0 0 1
90o z-axis -90o z-axis
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
??
Rick Parent - CIS681
Fixed Angles
Ordered triple of
rotations about global
axes, any triple can be
Y
used that doesn’t
immediately repeat an
axis, e.g., x-y-z, is fine,
so is x-y-x. But x-x-z is
not.
Z X
E.g., (qz, qy, qx)
Q = Rx(qx). Ry(qy). Rz(qz). P
Rick Parent - CIS681
Fixed Angles Using order z-y-x
Y Y
Z Z
X X
Orientation represented by
Original orientation (0,90,0)
Note: left-hand coordinate system
Rick Parent - CIS681
Fixed Angles
Using order z-y-x
Y Y
Z Z
X
X
Original (45,90,0)
Rick Parent - CIS681
Gimbal Lock
Using order z-y-x
From (0,90,0), how can the
object change its orientation?
Y What do these do?
a) (e,90,0)
b) (0,90+e,0)
Z
X c) (0,90,e)
Rick Parent - CIS681
Fixed Angles
(0,90,0) (-45,90,0)
Y Changing Y
Z-axis
parameter
Z Z
Is same as
X X-axis
X
rotation
(0,90,0) (0,90,45)
Rick Parent - CIS681
Fixed Angle Interpolation
(0,90,0) to (90,0,90)
(0,0,0)
(0,90,0) (90,0,90)
Rick Parent - CIS681
Euler Angles
Ordered triple of rotations about local axes,
As with fixed angles, any triple can be used that
doesn’t immediately repeat an axis, e.g., x-y-z, is
fine, so is x-y-x. But x-x-z is not.
Y y
Z
z x X
Rick Parent - CIS681
Euler Angles Use (z,y,x)
Show that Euler angle ordering Y y
is equivalent to reverse
ordering in fixed angles
P Rz (q1 )P
Z
P Rz (q1 )Ry (q 2 )Rz (q1 )P
P Rz (q1 )Ry (q 2 )Rz (q1 )Rz (q1 )P z x X
P Rz (q1 )Ry (q 2 )P
P Rz (q1 )Ry (q 2 )Rx (q 3 )Ry (q 2 )Rz (q1 )P
P Rz (q1 )Ry (q 2 )Rx (q 3 )Ry (q 2 )Rz (q1 )Rz (q1 )Ry (q 2 )P
P Rz (q1 )Ry (q 2 )Rx (q 3 )P
…and so has the same problems
Rick Parent - CIS681
Axis-Angle
Rotate object
(Ax,Ay,Az,q) by q around A
A
Y q
Z
X
Euler’s rotation theorem: An arbitrary rotation
may be described by only three parameters.
?
Rick Parent - CIS681
Axis-Angle Interpolation
1. Interpolate axis
A1 from A1 to A2 Rotate
q1 A axis about A1 x A2 to
Y get A
q
A1 x A2
A2
q2 2. Interpolate angle from
q1 to q2 to get q
Z
X
3. Rotate object
by q around A
Rick Parent - CIS681
Quaternions
Has the same information as
axis-angle but in a more
computational-friendly form
q =[s,v]=[s,x,y,z]
A
q
(cos(q/2),sin(q/2)*A)
Rick Parent - CIS681
Quaternions
Basic math operations
s1 v1 s2 v 2 s1 s2 v1 v 2
s1 v1 s2 v 2 s1s2 v1 v 2 s1v 2 s2v1 v1 v 2
q s2 x 2 y 2 z 2
q1 0 0 0 q
q1
s v
2
q
qq1 1 0 0 0
Rick Parent - CIS681
Quaternions - rotate a point
v = (x,y,z) => [0,v]
Rot q (v) v q0 v q1
Rick Parent - CIS681
Composite transformations
Rotation by p then by q is the same as rotation by qp
(where qp is quaternion q multiplied by quaternion p)
Rotq (Rotp (v)) Rotq ( p0 v p1 )
qp0 v p1q1
qp0 v (qp)1
Rotqp (v)
Rick Parent - CIS681
Quaternion Rotation
q
Unit quaternion =>
||q||
Rot s v Rot k s v Rot ks kv
Rot s v Rots v
Rick Parent - CIS681
Quaternion Interpolation
Fixed angles
(0,90,0) (90,0,90)
quaternions
[0.5,0.5,0.5,0.5]
[0.7,0.0,0.7,0.0]
Rick Parent - CIS681
Quaternion Interpolation
Linearly interpolating fixed Interpolating quaternions from
angles from (0,90,0) to (0.5,0.0,1.0,0.0) to (0.5,0.5,0.5,0.5)
(90,0,90) using sphereical linear interpolation
QuickTime™ an d a QuickTime™ an d a
Video d ecompressor Video d ecompressor
are need ed to see this p icture . are need ed to see this p icture .
Rick Parent - CIS681
