Dr. Scott Schaefer

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Quaternions and Complex Numbers

• Dr. Scott Schaefer

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers

• Defined by real and imaginary part
• where

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Complex Numbers and Rotations

• Given a point (x,y), rotate that point about the
  origin by

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Complex Numbers and Rotations

• Given a point (x,y), rotate that point about the
  origin by

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Complex Numbers and Rotations

• Given a point (x,y), rotate that point about the
  origin by

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Complex Numbers and Rotations

• Given a point (x,y), rotate that point about the
  origin by

Multiplication is rotation!!!
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Quaternions History

• Hamilton attempted to extend complex numbers from
  2D to 3D impossible
• 1843 Hamilton discovered
• a generalization to 4D and
• wrote it on the side of a
• bridge in Dublin
• One real part, 3 complex parts

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Quaternions
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Quaternions
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Quaternions
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Quaternions
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Quaternion Multiplication
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Quaternion Multiplication
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Quaternion Multiplication
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Quaternion Operations
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Quaternion Operations
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Quaternion Operations
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Quaternion Operations
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Quaternion Operations
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Quaternion Operations
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Quaternion Operations
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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

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Quaternions and Rotations

• Claim unit quaternions represent 3D rotation

Almost identical!!!
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Quaternions and Rotations

• The quaternion representing rotation about the
  unit axis v by is
• To convert to matrix, assume q(s,v) and q1

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Quaternions and Rotations

• The quaternion representing rotation about the
  unit axis v by is
• To convert to matrix, assume q(s,v) and q1

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Quaternions vs. Matrices

• Quaternions take less space (4 numbers vs. 9 for
  matrices)
• Rotating a vector requires 28 multiplications
  using quaternions vs. 9 for matrices
• Composing to rotations using quaternions q1q2
  requires 16 multiples vs. 27 for matrices
• Quaternions are typically not hardware
  accelerated whereas matrices are

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Quaternions and Interpolation

• Given two orientations q1 and q2, find the
  orientation halfway between

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Quaternions and Interpolation

• Given two orientations q1 and q2, find the
  orientation halfway between

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Quaternions and Interpolation

• Unit quaternions represent points on a 4D
  hyper-sphere
• Interpolation on the sphere gives rotations that
  bend the least

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Quaternions and Interpolation

• Unit quaternions represent points on a 4D
  hyper-sphere
• Interpolation on the sphere gives rotations that
  bend the least

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Quaternions and Interpolation

• Unit quaternions represent points on a 4D
  hyper-sphere
• Interpolation on the sphere gives rotations that
  bend the least
• May need to interpolate between q1 and q2

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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Quaternion Interpolation
Euler Angle Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Quaternions and Interpolation
Euler Angle Interpolation
Quaternion Interpolation
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Exponential Forms

• Eulers formula

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Quaternions in Exponential Form