3D orientation
• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map
Joints and rotations
Rotational DOFs are widely used in character animation
3 translational DOFs
48 rotational DOFs
Each joint can have up to 3 DOFs
1 DOF: knee
2 DOF: wrist
3 DOF: arm
Representation of orientation
• Homogeneous coordinates (review)
• 4X4 matrix used to represent translation,
scaling, and rotation
• a point in the space is represented as
• Treat all transformations the same so that
they can be easily combined
Translation
new point
translation
matrix
old point
Scaling
new point
scaling matrix
old point
Rotation
X axis
Y axis
Z axis
Composite transformations
A series of transformations on an
object can be applied as a series of
matrix multiplications
: position in the global coordinate
: position in the local coordinate
Interpolation
• In order to “move things”, we need both
translation and rotation
• Interpolating the translation is easy, but what
about rotations?
Motivation
• Finding the most natural and compact way to
present rotation and orientations
• Orientation interpolation which result in a natural
motion
• A closed mathematical form that deals with
rotation and orientations (expansion for the
complex numbers)
Rotation Matrix
• A general rotation can be represented by a single
3x3 matrix
• Length Preserving (Isometric)
• Reflection Preserving
• Orthonormal
 ux

R   vx
w
 x
uy
vy
wy
uz 

vz 
wz 
Fixed Angle Representation
• Angles used to rotate about fixed axes
• Orientations are specified by a set of 3 ordered
parameters that represent 3 ordered rotations
about fixed axes, ie. first about x, then y, then z
• Many possible orderings, don’t have to use all 3
axes, but can’t do the same axis back to back
Fixed Angle Representation
• A rotation of 10,45, 90 would be written as
• Rz(90) Ry(45), Rx(10) since we want to first
rotate about x, y, z. It would be applied then to
the point P…. RzRyRx P
• Problem occurs when two of the axes of rotation
line up on top of each other. This is called “Gimbal
Lock”
Gimbal Lock
• A 90 degree rotation about the y axis
essentially makes the first axis of rotation align
with the third.
• Incremental changes in x,z produce the same
results – you’ve lost a degree of freedom
Gimbal Lock
• Phenomenon of two
rotational axis of an object
pointing in the same
direction.
• Simply put, it means your
object won't rotate how you
think it ought to rotate.
Interpolation of orientation
• How about interpolating each entry of the
rotation matrix?
• The interpolated matrix might no longer be
orthonormal, leading to nonsense for the inbetween rotations
Interpolation of orientation
Example: interpolate linearly from a positive 90
degree rotation about y axis to a negative 90
degree rotation about y
Linearly interpolate each
component and halfway
between, you get this...
Properties of rotation matrix
• Easily composed?
• Interpolate?
• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map
Fixed angle
• Angles used to rotate about fixed axes
• Orientations are specified by a set of 3 ordered
parameters that represent 3 ordered rotations
about fixed axes
• Many possible orderings
Fixed angle
• A rotation of Rz(90)Ry(60)Rx(30) looks like
Euler angle
• Same as fixed angles, except now the axes
move with the object
• An Euler angle is a rotation about a single
Cartesian axis
• Create multi-DOF rotations by concatenating
Euler angles
• evaluate each axis independently in a set order
Euler angle vs. fixed angle
• Rz(90)Ry(60)Rx(30) = Ex(30)Ey(60)Ez(90)
• Euler angle rotations about moving axes written
in reverse order are the same as the fixed axis
rotations
Z
Y
X
Gimbal Lock (again!)
• Rotation by 90o causes a loss of a degree of freedom
1
x
z
z
z
3
/2
1
y
y
y
x
1
x’
x
1
x’
Gimbal Lock
• Phenomenon of two
rotational axis of an object
pointing in the same
direction.
• Simply put, it means your
object won't rotate how you
think it ought to rotate.
Gimbal Lock
A Gimbal is a hardware implementation of
Euler angles used for mounting
gyroscopes or expensive globes
Gimbal lock is a basic problem with
representing 3D rotation using Euler
angles or fixed angles
Gimbal lock
When two rotational axis of an object point in the same
direction, the rotation ends up losing one degree of
freedom
Properties of Euler angle
• Easily composed?
• Interpolate?
• How about joint limit?
• What seems to be the problem?
Euler Angles
• A general rotation is a combination of three
elementary rotations: around the x-axis (x-roll) ,
around the y-axis (y-roll) and around the z-axis
(z-roll).
Euler
Angles
Euler Angles and Rotation
Matrices
0
0
1

 0 cos 1 sin 1
x  roll (1 )  
0  sin 1 cos 1

0
0
0

 cos  3

  sin  3
z  roll ( 3 )  
0

 0

sin  3
cos  3
c2 c3


 s1s2 c3  c1s3
R(1 , 2 , 3 )  
cs c s s
 1 2 3 1 3

0

0
0
0

0
0

1 
 cos  2

 0
y - roll(  2 )  
sin  2

 0

0 0

0 0
1 0

0 1 
c2 s3
 s2
s1 s2 s3  c1c3
s1c2
c1s2 s3  s1c3
c1c2
0
0
0

0
0

1 
0  sin  2
1
0
0
cos  2
0
0
0

0
0

1 
Euler angles interpolation
y
π
x
z
y
π
y
x
z
x
x-roll
π
z
y
y-roll
π
x
π
R(0,0,0),…,R(t,0,0),…,R(,0,0)
t[0,1]
y
z-roll
π
z
R(0,0,0),…,R(0,t, t),…,R(0,, )
x
z
Euler Angles Interpolation
Unnatural movement !
• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map
Goal
• Find a parameterization in which
• a simple steady rotation exists between two key
orientations
• moves are independent of the choice of the
coordinate system
Axis angle
• Represent orientation as a vector and a scalar
• vector is the axis to rotate about
• scalar is the angle to rotate by
Angle and Axis
• Any orientation can be represented by a 4-tuple
• angle, vector(x,y,z) where the angle is the amount to
rotate by and the vector is the axis to rotate about
• Can interpolate the angle and axis separately
• No gimbal lock problems!
• But, can’t efficiently compose rotations…must convert to
matrices first!
Angular displacement
v||  (n  v ) n
v  v  v||
n
R[v ]  v cos   ( n  v ) sin 
 v cos   ( n  v) sin 
R[v|| ]  v||
v
v||
• (,n) defines an angular displacement
of  about an axis n

n  v
R[v]  R[v||  v ]  R[v|| ]  R[v ]  v||  v cos  (n  v) sin 
 (n  v)n  (v  (n  v)n) cos  (n  v) sin 
 v cos  n(n  v)(1  cos )  (n  v) sin 
v
R[v ]
Properties of axis angle
• Can avoid Gimbal lock. Why?
• Can interpolate the vector and the scalar
separately. How?
Axis angle interpolation
• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map
Quaternion
4 tuple of real numbers:
scalar
vector
Same information as axis angles but in a different form
•
Quaternions
?
Extend the concept of rotation in 3D to 4D.
• Avoids the problem of "gimbal-lock" and allows for the
implementation of smooth and continuous rotation.
• In effect, they may be considered to add a additional
rotation angle to spherical coordinates ie. Longitude,
Latitude and Rotation angles
• A Quaternion is defined using four floating point values
|x y z w|. These are calculated from the combination of
the three coordinates of the rotation axis and the
rotation angle.
How do quaternions relate to 3D
• Solution to "Gimbal
lock"
animation?
• Instead of rotating an object through a series of
successive rotations, a quaternion allows the
programmer to rotate an object through a single arbitary
rotation axis.
• Because the rotation axis is specifed as a unit direction
vector, it may be calculated through vector mathematics
or from spherical coordinates ie (longitude/latitude).
• Quaternions interpolation : smooth and predictable
rotation effects.
Quaternion to Rotation Matrix
Q  X Y
1  2Y  2Z

M   2 XY  2ZW
 2 XZ  2YW

2
2
Z W
2 XY  2ZW
1  2 X  2Z
2
2YZ  2 XW
2
2 XZ  2YW 

2YZ  2 XW 
2
2
1  2 X  2Y 
Quaternions Definition
• Extension of complex numbers
• 4-tuple of real numbers
• s,x,y,z or [s,v]
• s is a scalar
• v is a vector
• Same information as axis/angle but in a different
form
• Can be viewed as an original orientation or a
rotation to apply to an object
Quaternions Math
Quaternions properties
• The conjugate and
magnitude are similar to
complex numbers
q  ( s , v )
q  q * q  s 2  v x2  v y2  vz2
• Quaternions are non commutative
q1 = (s1,v1) q2 = (s2,v2)
q1*q2 = (s1s2 – v1.v2 , s1v2 + s2v1 + v1 x v2)
q
• inverse: q 
q*q
• unit quaternion:
q  1  q 1  q
1
Quaternion Rotation
To rotate a vector, v using quaternion math
• represent the vector as [0,v]
• represent the rotation as a quaternion, q
q  Rot , x , y , z   cos 2, sin  2  x, y, z 
v  Rot v   q  v  q
1
Quaternions as Rotations
• Rotation of P=(0,r) about the unit vector n by an angle
θ using the unit quaternion q=(s,v)
Rq [ P]  qPq1  (0, (s 2  v  v)r  2v(v  r )  2sv  r )
but q=(cos½θ, sin½θ•n) where |n|=1
Rq [ P]  (0 , (cos 2  2  sin 2  2)r  2n(n  r ) sin 2  2 
2(n  r ) cos 2 sin  2 )
 (0 , cos r  (1  cos ) n (n  r )  (n  r ) sin  )
Quaternions as Rotations
• Concatenating rotations – rotate using q1 and
then using q2 is like rotation using q2*q1
q2 * (q1 * P * q11 ) * q21  (q2 * q1 ) * P * (q11 * q21 )
 (q2 * q1 ) * P * (q2 * q1 ) 1
Quaternion math
Unit quaternion
Multiplication
Quaternion math
Conjugate
Inverse
the unit length quaternion
Quaternion Rotation
If
is a unit quaternion and
then
results in
rotating about
proof: see Quaternions by Shoemaker
by
Quaternion Rotation
Quaternion composition
If
and
are unit quaternion
the combined rotation of first rotating by
then by
is equivalent to
and
Matrix form
Quaternion interpolation
1-angle rotation can be
represented by a unit circle
2-angle rotation can be
represented by a unit sphere
• Interpolation means moving on n-D sphere
• Now imagine a 4-D sphere for 3-angle rotation
Quaternion interpolation
• Moving between two points on the 4D unit
sphere
• a unit quaternion at each step - another point
on the 4D unit sphere
• move with constant angular velocity along the
great circle between the two points on the 4D
unit sphere
Quaternion interpolation
Direct linear interpolation does not work
Linearly interpolated intermediate points are not
uniformly spaced when projected onto the circle
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion
Rotations in Reality
• It’s easiest to express rotations in Euler angles or
Axis/angle
• We can convert to/from any of these representations
• Choose the best representation for the task
• input:Euler angles
• interpolation: quaternions
• composing rotations: quaternions, orientation matrix
• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map
Exponential map
• Represent orientation as a vector
• direction of the vector is the axis to rotate about
• magnitude of the vector is the angle to rotate by
• Zero vector represents the identity rotation
Properties of exponential map
• No need to re-normalize the parameters
• Fewer DOFs
• Good interpolation behavior
• Singularities exist but can be avoided
Choose a representation
• Choose the best representation for the task
• input:
Euler angles
• joint limits:
Euler angles, quaternion (harder)
• interpolation:
quaternion or exponential map
• compositing:
quaternions or orientation matrix
• rendering:
orientation matrix ( quaternion can be
represented as matrix as well)