Quaternions

1
Quaternions

• CS 4451

2
Euler Angles
Rotations are defined as rotations about the x, y
and z axes. Rotations about any other vector in
space have to be broken down into equivalent
rotations about the major axes.
y
z
x
3
Animation

y
To produce the effect of a smooth rotation about
some arbitrary axis in space, we generate a
series of FRAMES.
z
x
4
Frames

y
With Euler Angles, knowing the transformations
for the entire rotation does not help us with the
intermediate rotations. Each is a unique set of
three rotations about the x,y and z axes
z
x
5
Quaternions

• Invented in 1800s
• Alternate way to describe rotations about an
  arbitrary axis.
• Useful in animation because it is easy to
  interpolate rotations
• 4-tuple described by four real numbers q w, x,
  y, z

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Quaternion Operations

• q w,x,y,z where w is a scaler related to an
  angle and x,y,z, is a vector.
• We can write this as ?, v where
• w and v x,y,z
• ?1, v1 ?2, v2 ?1 ?2, v1 v2
• ?1, v1 ?2, v2
• (?1?2 - v1v2), (?1v2 ?2 v1 v1 x v2)

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Quaternion Operations

• q ?(w2 x2 y2 z2)
• q-1 ?, -v / q2
• qq-1 1, 0, 0, 0

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Describing Motion in Animations

• Only consider quaternions such that q 1
• To perform a rotation of point d about the unit
  vector v of ? degrees we compute
• q-10,dq (remember q ?, v)
• The result will be a vector that represents a
  rotation of point d about the vector v with angle
  ? given by the equation ? cos(?/2)

9
Example
Rotation of 180 degrees about the vector 1,0,1,
of point (1,0,0).
y
(0,0,1)
z
x
(1,0,0)
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Notation for this example

• What is q ?, v such that q 1?
• cos (180/2) cos(90) 0
• v ?2//2, 0, ?2//2
• q 0, ?2//2, 0, ?2//2
• q-1 ?, -v /q2 0, -?2//2, 0, -?2//2
• 0,d 0, 1, 0, 0

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To Rotate

• ?1, v1 ?2, v2
• (?1?2 - v1v2), (?1v2 ?2 v1 v1 x v2)
• q-10,dq
• 0, ?2//2, 0, ?2//20, 1, 0, 00, -?2//2, 0,
  -?2//2
• 0, 0, 0 1, new rotated coordinate is (0,0,1)

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Why is this cool?
Rotation of 90 degrees about the vector 1,0,1,
of point (1,0,0).
y
z
x
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Why is this cool?

• Finding the quaternion representation for all the
  intermediate frame rotations is easy.
• For example, a rotation of 90 degrees of the
  point (1,0,0) about the vector 1,0,1
• What is q ?, v such that q 1?
• cos (90/2) cos(45) ?2//2
• v 1/2, 0, 1/2
• q45 ?2//2, 1/2, 0, 1/2

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How to find the unit quaternion?

• Assume that you want to rotate 30 degrees about
  the axis 0, 3, 4
• First find unit vector 0,3,4/5 0,0.6, 0.8
• Unit quaternion is
• cos (15), 0sin(15), 0.6sin(15), 0.8sin(15)
• 0.966, 0, 0.60.259, 0.80.259
• 0.966, 0, 0.155, 0.207