Maths - Axis-Angle - Sample Orientations

In order to try to explain things and give some examples we can try I thought it might help to show the rotations for a finite subset of the rotation group. We will use the set of rotations of a cube onto itself, this is a permutation group which gives 24 possible rotations as explaned on this page.

Heading applied first giving 4 possible orientations:

 

aeroplane

reference orientation

angle = 0 degrees
axis = 1,0,0

aeroplane

rotate by 90 degrees about y axis

angle = 90 degrees
axis = 0,1,0

aeroplane

rotate by 180 degrees about y axis

angle = 180 degrees
axis = 0,1,0

aeroplane

rotate by 270 degrees about y axis

angle = 90 degrees
axis = 0,-1,0

or

angle = -90 degrees
axis = 0,1,0

Then apply attitude +90 degrees for each of the above: (note: that if we went on to apply bank to these it would just rotate between these values, the straight up and straight down orientations are known as singularities because they can be fully defined without using the bank value)

aeroplane

angle = 90 degrees
axis = 0,0,1

aeroplane

angle = 120 degrees
axis = 0.5774,0.5774,0.5774

aeroplane

angle = 180 degrees
axis = 0.7071,0.7071,0

aeroplane

angle = 120 degrees
axis = -0.5774,-0.5774,0.5774

Or instead apply attitude -90 degrees (also a singularity):

aeroplane

angle = 90 degrees
axis = 0,0,-1

(equivalent rotation to:
angle = -90 degrees
axis = 0,0,1)

aeroplane

angle = 120 degrees
axis = -0.5774,0.5774,-0.5774

aeroplane

angle = 180 degrees
axis = -0.7071,0.7071,0

aeroplane

angle = 120 degrees
axis = 0.5774,-0.5774,-0.5774

Normally we don't go beyond attitude + or - 90 degrees because these are singularities, instead apply bank +90 degrees:

aeroplane

angle = 90 degrees
axis = 1,0,0

aeroplane

angle = 120 degrees
axis = 0.5774,0.5774,-0.5774

aeroplane

angle = 180 degrees
axis = 0,0.7071,-0.7071

aeroplane

angle = 120 degrees
axis = 0.5774,-0.5774,0.5774

Apply bank +180 degrees:

aeroplane

angle = 180 degrees
axis = 1,0,0

aeroplane

angle = 180 degrees
axis = 0.7071,0,-0.7071

aeroplane

angle = 180 degrees
axis = 0,0,1

aeroplane

angle = 180 degrees
axis = 0.7071,0,0.7071

Apply bank -90 degrees:

aeroplane

angle = 90 degrees
axis = -1,0,0

(equivalent rotation to:
angle = -90 degrees
axis = 1,0,0)

aeroplane

angle = 120 degrees
axis = -0.5774,0.5774,0.5774

aeroplane

angle = 180 degrees
axis = 0,0.7071,0.7071

 

aeroplane

angle = 120 degrees
axis = -0.5774,-0.5774,-0.5774

encoding of these rotations in quaternions is shown here.
encoding of these rotations in matrices is shown here.
encoding of these rotations in euler angles is shown here.


metadata block
see also:

 

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