Euler vs Hamilton
by
Danielle Smith
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Euler vs Hamilton Daniel Smith && Jan CW Kroeze University of Pretoria
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INTRO
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Various ways to rotate stuff
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Various ways to rotate stuff Quaternions
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Various ways to rotate stuff Quaternions Pitch-Yaw-Roll
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Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices
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Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices Angle-Axis
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Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices Angle-Axis WHICH ONE??
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Pitch-Yaw-Roll is just bad (Gimbal Lock)
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Angle-Axis is complicated
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Quaternions are popular
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Rotation Matrices ... exist
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How do we choose?
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METRICS
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SPEED
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Rotating is a lot of FP math
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sin, cos and FP multiplication
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We measure total ns spent on calculations
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STABILITY
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What is stability?
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Rounding errors! 4.0 + 3.0 = 6.99999999
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Rotations get a bit wonky over time
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Orthonormalisation
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(Orthogonal + normal) + isation
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Orthogonalisation + normalisation
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Un-wonky rotations
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How do you measure stability?
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Compare limited-precision with perfect precision
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(no such thing as perfect precision)
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Compromise: double is “more perfect” than single
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Discrepancy between single- and double-precision
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EXPERIMENT
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Implement a 3D camera
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Rotate the camera using different methods
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Measure stability and speed
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RESULTS
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Instability / Time after about a minute of input
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Total CPU Time
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Rotation Matrices accumulate instability at a much faster rate
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(unless they’re orthonormalised)
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Instability / Time
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Quaternions are very well behaved
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Orthonormalising quaternions is a silly idea
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Quaternions 3x faster than Rotation Matrices
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3x faster!! (Order now, 50% off!)
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(in the order of nanoseconds, so what difference does it make?)
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Orthonormalising Quaternions make them faster?
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???
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CONCLUSION
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What does it mean?
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Keep using quaternions!
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(but perhaps not for the reasons you expect)
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SPEED
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STABILITY
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INTERPOLATION
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SUNROOF!
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