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Rick =
Parent -
CIS681
<=
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![](/://3/3D"master03_background.gif")
Rick =
Parent
- CIS681
ORIENTATION
=
In=
terpolating
rotations is difficult
Rick Parent - CIS681
Object Representation=
=
Desired
operations
Interpolation
between transformations
Concatenation
of one transformation after another
Define object in world space
Object space data
Scale
Rotation
Translation
Handle
scale, rotation, translation, independently
Rotation
deserves special attention!
![](/://3/3D"slide0099_image001.gif")
Rick Parent - CIS681
Repeated Rotations: Error Management
Task: Rotate an ob=
ject
some Dq every fram=
e
Issue: Avoiding
accumulated roundoff error
Rick Parent - CIS681
Repeated Rotations: Error Management
Method
2
D
=/://3/3D create_rotation_matrix(Dq)
M
create_rotation_matrix(q)
M
=/://3/3D D M
Object
=/://3/3D apply M to object
Method
1
M
=/://3/3D create_rotation_matrix(Dq)
Object
=/://3/3D apply M to Object
=
Method
3
q =/://3/3D q + Dq&=
#13;
M
=/://3/3D create_rotation_matrix(q=
)
Object
=/://3/3D apply M to object
![](/://3/3D"slide0101_image002.gif")
<=/://3/3D repeat
![](/://3/3D"slide0101_image003.gif")
<=/://3/3D repeat
![](/://3/3D"slide0101_image004.gif")
<=/://3/3D repeat
Rick Parent - CIS681
![](/://3/3D"slide0102_image005.gif")
Orientation Representation
![](/://3/3D"slide0102_image006.gif")
![](/://3/3D"slide0102_image007.gif")
![](/://3/3D"slide0102_image008.gif")
![](/://3/3D"slide0102_image009.gif")
![](/://3/3D"slide0102_image010.gif")
![](/://3/3D"slide0102_image011.gif")
![](/://3/3D"slide0102_image012.gif")
orientation
Rick Parent - CIS681
![](/://3/3D"slide0103_image013.gif")
Interpolation
![](/://3/3D"slide0103_image014.gif")
![](/://3/3D"slide0103_image015.gif")
![](/://3/3D"slide0103_image016.gif")
![](/://3/3D"slide0103_image017.gif")
O1
![](/://3/3D"slide0103_image018.gif")
O2
![](/://3/3D"slide0103_image019.gif")
O 1.5
Rick Parent - CIS681
![](/://3/3D"slide0104_image020.gif")
Concatenation
![](/://3/3D"slide0104_image021.gif")
![](/://3/3D"slide0104_image022.gif")
![](/://3/3D"slide0104_image023.gif")
O1
![](/://3/3D"slide0104_image024.gif")
O2
Rick Parent - CIS681
Orientation Representation
Rotation Matrix
Fixed Angles
Euler Angles
Axis-Angle
Quaternion
Rick Parent - CIS681
![](/://3/3D"slide0106_image025.gif")
Rotation Matrices
0
0
0
1
a
d
g
b
c
e
f
h
i
0
0
0
9 values but 3 degrees of freedom
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Euler’s rotation theorem: An arbitrary rotatio=
n may be described by =
only
three parameters.
Rick Parent - CIS681
![](/://3/3D"slide0107_image026.gif")
Rotation Matrices
0
0
0=
1
0
1
0
-1
0
0
0
0
1
0
0
0
0
0
0=
1
0
-1
0
1
0
0
0
0
1
0
0
0
Can’t interpolate =
rotation
matrices
90o z-axis
-90o z-axis
![](/://3/3D"slide0107_image027.gif")
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
??
Rick Parent - CIS681
![](/://3/3D"slide0108_image030.gif")
Fixed Angles
E.g., (qz, qy, qx)
X
Z
Y
Q =/://3/3D Rx(qx). Ry (qy). Rz (qz). P
Ordered triple of =
rotations
about global axes, 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.
=
Rick Parent - CIS681
![](/://3/3D"slide0109_image031.gif")
Fixed Angles
X
Z
Y
X
Z
Y
Note: left-hand coordinate system
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Origin=
al
orientation
Orientation represented by (0,90,0)
Using order z-y-x
Rick Parent - CIS681
![](/://3/3D"slide0110_image032.gif")
Fixed Angles
(45,90,0)
X
Z
Y
X
Z
Y
Origina=
l
Using order z-y-x
Rick Parent - CIS681
![](/://3/3D"slide0111_image033.gif")
Gimbal Lock
From (0,90,0), how can th=
e object
change its orientation?
X
Z
Y
=
a) (e,90,0)
=
b) (0,90+e,0)
=
c) (0,90,e)
What do these do?
Using order z-y-x
![](/://3/3D"slide0111_image034.gif")
Rick Parent - CIS681
![](/://3/3D"slide0112_image035.gif")
Fixed Angles
(-45,90,0)
(0,90,0)
X
Z
Y
Z
X
Y
![](/://3/3D"slide0112_image036.gif")
Is
same as
X-axis
rotation
(0,90,45)
(0,90,0)
Changing <=
span
style=/://3/3D'position:absolute;top:44.5%;left:45.31%;width:17.22%;height:5.25%=
'>Z-axis
parameter
Rick Parent - CIS681
Fixed Angle Interpolation
(0,90,0) to (90,0,90)
![](/://3/3D"slide0113_image037.jpg")
![](/://3/3D"slide0113_image041.gif")
![](/://3/3D"slide0113_image043.jpg")
(0,0=
,0)
(90=
,0,90)
(0,90=
,0)
![](/://3/3D"slide0113_image045.jpg")
Rick Parent - CIS681
![](/://3/3D"slide0114_image047.gif")
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.X
Y
Z
x
y
z
Rick Parent - CIS681
![](/://3/3D"slide0115_image049.gif")
Euler Angles
X
Y
Z
x
y
z
Use (z,y,x)
![](/://3/3D"slide0115_image052.gif")
=
Show that Euler angle ordering =
is equivalent to reverse ordering in fixed angles
…and so has the same problems=
Rick Parent - CIS681
![](/://3/3D"slide0116_image053.gif")
Axis-Angle
![](/://3/3D"slide0116_image054.gif")
A=
q=
(Ax,A<=
span
style=/://3/3D'font-size:67%;position:relative;top:.37em;mso-text-raise:-25%'>y,Az ,q)
Euler’s rotation theorem: An arbitrary rotatio=
n may be described by =
only
three parameters.
X
Y
Z
Rotate object by
q around A
?
Rick Parent - CIS681
![](/://3/3D"slide0117_image055.gif")
Axis-Angle Interpolation
![](/://3/3D"slide0117_image056.gif")
A=
1
q2=
1.
Interpolate axis from A1 to A2 Rotate <=
b>axis
about A1 x A2 to get A
![](/://3/3D"slide0117_image057.gif")
X
Y
Z
A2=
q1=
2.
Interpolate angle from =
q1
to q2 to get =
q
![](/://3/3D"slide0117_image058.gif")
3.
Rotate object by <=
b>q
around A
A1 x A2
![](/://3/3D"slide0117_image059.gif")
A
![](/://3/3D"slide0117_image060.gif")
q
Rick Parent - CIS681
![](/://3/3D"slide0118_image061.gif")
Quaternions
![](/://3/3D"slide0118_image062.gif")
A=
q=
(cos(q/2),sin(<=
b>q/2)*A)
q =/://3/3D[s,v]
=/://3/3D[s,x,y,z]
Has
the same information as axis-angle
but in a more computa=
tional-friendly
form
Rick Parent - CIS681
Quaternions
![](/://3/3D"slide0119_image065.gif")
Basic math operations
Rick Parent - CIS681
Quaternions - rotate a point
v =/://3/3D (x,y,z) =/://3/3D> [0,v]
![](/://3/3D"slide0120_image068.gif")
Rick Parent - CIS681
Composite transformations
![](/://3/3D"slide0121_image071.gif")
Rotation by p then by q is the same as rot=
ation
by qp
(where=
qp is
quaternion q multiplied by quaternion p)
Rick Parent - CIS681
![](/://3/3D"slide0122_image072.gif")
Quaternion
Rotation
q
||q||
Unit quaternion =/://3/3D>
![](/://3/3D"slide0122_image075.gif")
![](/://3/3D"slide0122_image078.gif")
Rick Parent - CIS681
Quaternion Interpolation
Fixed
angles
![](/://3/3D"slide0123_image079.jpg")
![](/://3/3D"slide0123_image081.jpg")
![](/://3/3D"slide0123_image083.jpg")
![](/://3/3D"slide0123_image085.jpg")
![](/://3/3D"slide0123_image086.gif")
![](/://3/3D"slide0123_image087.jpg")
![](/://3/3D"slide0123_image088.jpg")
(90,0,90)
(0,90,0)
qua=
ternions
![](/://3/3D"slide0123_image089.jpg")
[0.7=
,0.0,0.7,0.0]
[0.=
5,0.5,0.5,0.5]
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The axis is really only 2 DoFs -=
its
length is irrelevant
' + ENDSHOW_MESG + '
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