Quaternions

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Quaternions
Visualization and Animation Course

• Presented by
• Zachi Karni
• Tali Sapir

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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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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 (cont)
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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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Euler Angles and Rotation Matrices
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Gimbal Lock

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

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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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Goal

• Find a parametrization 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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Angular displacement

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

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Quaternions Definition

• Quaternions are an extension of complex numbers

q s vxi vyj vzk or q (s,v) where
i2 j2 k2 -1 ij k and ji -k jk i
and kj -i ki j and ik -j
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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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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
same form as angular displacement !
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Quaternions as Rotations cont.
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Interpolating using Quaternions

• The animator sets a sequence of key orientations
• The mission interpolate between them

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Interpolating using quaternions

• Rotations are represented by unit quaternions
  therefore the group of rotations lies on a 4D
  unit hypersphere

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Interpolating two quaternions

• Linear interpolation move along a straight line
• Spherical linear interpolation move along an arc

q(u) ?(u)q1?(u)q2 for u?0,1 Solve the
following equations to get ?(u) and ?(u)
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Interpolating two quaternions

• Spherical linear interpolation

• Moving on the arc from p to q has the same effect
  as moving on the arc from p to q.
• Choose the shorter path.

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Interpolating a sequence of quaternions

• Spherical linear interpolation between more than
  two key orientations causes non smooth motion
  because of derivatives discontinuities at the
  keys
• We need the spherical equivalent for cubic spline
  in 4D

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linear interpolation
cubic spline interpolation