3D orientation

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3D orientation
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• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map

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Joints and rotations
Rotational DOFs are widely used in character
animation
Each joint can have up to 3 DOFs
1 DOF knee
2 DOF wrist
3 DOF arm
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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

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Translation
translation matrix
new point
old point
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Scaling
scaling matrix
new point
old point
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Rotation
X axis
Y axis
Z axis
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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
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Interpolation

• In order to move things, we need both
  translation and rotation
• Interpolating the translation is easy, but what
  about rotations?

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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)

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Rotation Matrix

• A general rotation can be represented by a single
  3x3 matrix
• Length Preserving (Isometric)
• Reflection Preserving
• Orthonormal

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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, dont have to use all 3
  axes, but cant do the same axis back to back

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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

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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 youve lost a degree of freedom

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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.

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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
  in-between rotations

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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...
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Properties of rotation matrix

• Easily composed?
• Interpolate?

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• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map

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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

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Fixed angle

• A rotation of Rz(90)Ry(60)Rx(30) looks like

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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

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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

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Gimbal Lock (again!)

• Rotation by 90o causes a loss of a degree of
  freedom

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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.

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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
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Gimbal lock
When two rotational axis of an object point in
the same direction, the rotation ends up losing
one degree of freedom
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Properties of Euler angle

• Easily composed?
• Interpolate?
• How about joint limit?
• What seems to be the problem?

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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).

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Euler Angles
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Euler Angles and Rotation Matrices
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Euler angles interpolation
R(0,0,0),,R(?t,0,0),,R(?,0,0) t?0,1
R(0,0,0),,R(0,?t, ?t),,R(0,?, ?)
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Euler Angles Interpolation ?Unnatural movement !
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• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map

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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

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Axis angle

• Represent orientation as a vector and a scalar
• vector is the axis to rotate about
• scalar is the angle to rotate by

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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, cant efficiently compose rotationsmust
  convert to matrices first!

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Angular displacement

• (?,n) defines an angular displacement of ? about
  an axis n

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Properties of axis angle

• Can avoid Gimbal lock. Why?
• Can interpolate the vector and the scalar
  separately. How?

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Axis angle interpolation
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• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map

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Quaternion
4 tuple of real numbers
scalar
vector
Same information as axis angles but in a
different form
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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.

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How do quaternions relate to 3D animation?

• Solution to "Gimbal lock"
• 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.

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Quaternion to Rotation Matrix
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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

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Quaternions Math
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Quaternions properties

• The conjugate and magnitude are similar to
  complex numbers

• Quaternions are non commutative

q1 (s1,v1) q2 (s2,v2) q1q2 (s1s2 v1.v2
, s1v2 s2v1 v1 x v2)

• inverse
• unit quaternion

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Quaternion Rotation

• To rotate a vector, v using quaternion math
• represent the vector as 0,v
• represent the rotation as a quaternion, q

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Quaternions as Rotations

• Rotation of P(0,r) about the unit vector n by an
  angle ? using the unit quaternion q(s,v)

but q(cos½?, sin½?n) where n1
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Quaternions as Rotations
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Quaternion math
Unit quaternion
Multiplication
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Quaternion math
Conjugate
Inverse
the unit length quaternion
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Quaternion Rotation
proof see Quaternions by Shoemaker
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Quaternion Rotation
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Quaternion composition
If and are unit quaternion
the combined rotation of first rotating by
and then by is equivalent to
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Matrix form
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Quaternion interpolation
1-angle rotation can be represented by a unit
circle

• Interpolation means moving on n-D sphere
• Now imagine a 4-D sphere for 3-angle rotation

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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

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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
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Rotations in Reality

• Its 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
• inputEuler angles
• interpolation quaternions
• composing rotations quaternions, orientation
  matrix

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• Rotation matrix
• Fixed angle and Euler angle
• Axis angle
• Quaternion
• Exponential map

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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

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Properties of exponential map

• No need to re-normalize the parameters
• Fewer DOFs
• Good interpolation behavior
• Singularities exist but can be avoided

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Choose a representation

• Choose the best representation for the task
• input
• joint limits
• interpolation
• compositing
• rendering

Euler angles
Euler angles, quaternion (harder)
quaternion or exponential map
quaternions or orientation matrix
orientation matrix ( quaternion can be
represented as matrix as well)