Quaternionic%20Splines%20of%20Camera%20Paths

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Quaternionic Splines of Paths

• Robert Shuttleworth
• Youngstown State University
• Professor George Francis, Director
• illiMath2001

NSF VIGRE REU UIUC-NCSA
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Order of Events

• History of the quaternions
• What is a quaternion?
• Significance to Computer Graphics
• Splining of Paths
• RTICA

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History of the Quaternions

• Sir William Rowan Hamilton (1805-1865)

• Royal Canal, Dublin October 16, 1843

• First example of a Lie Group

• Gibbs vector dot and cross product

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What is a quaternion?
Generalizations of the complex numbers into 4D
i2 j2 k2 ijk -1
Complex Numbers (C) Quaternions (H)
Multiplication of quaternions is not normally
commutative!
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Rotation Matrices
In 2D
Rotation matrices are not optimal!
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What is SO(3)?

• Orthogonal UTU-1
• SO(n) special orthogonal group
• SO(2) rotations about the origin in 2D
• SO(3) set of rotations in 3D

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Rotations with Quaternions
S3 in R4 is a Lie Group under Quaternionic
Multiplication
In R3, p qpq-1
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Advantages of Quaternions in Computer Graphics

• Coordinate system independent
• Easy to represent rotations
• Less values need to be stored when compared to
  matrices
• Allows efficient splining of paths

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Linear Interpolation (LERP)
Note Seven numbers are needed to perform this
calculation
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Spherical Linear Interpolation (SLERP)
where
d acos (A.B)
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Geometry of SLERP in the Plane
A
B
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SLERP with Three Points
B
L1(t)
K
A
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RTICA
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Any Questions?