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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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