Maths - Axis Angle examples 90 degree steps

Maths - Axis-Angle - Sample Orientations

In order to try to explain things I thought it might help to work out a simple case where rotations are only allowed in multiples of 90 degrees. This should make it easier to illustrate the orientation with a simple aeroplane figure, we can rotate this either about the x,y or z axis as shown here:

When we combine these rotations about the x,y and z axis in 90 degree multiples there are 24 possible orientations as shown here:

heading applied first giving 4 possible orientations:

reference orientation

angle = 0 degrees
axis = 1,0,0

rotate by 90 degrees about y axis

angle = 90 degrees
axis = 0,1,0

rotate by 180 degrees about y axis

angle = 180 degrees
axis = 0,1,0

rotate by 270 degrees about y axis

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

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)

angle = 90 degrees
axis = 0,0,1

aeroplane

angle = 120 degrees
axis = 0.5774,0.5774,0.5774

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):

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

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

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

aeroplane

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

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

Apply bank +180 degrees:

angle = 180 degrees
axis = 1,0,0

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

angle = 180 degrees
axis = 0,0,1

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)

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

aeroplane

angle = 180 degrees
axis = 0,0.7071,0.7071

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.

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see also:

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Book Shop - Further reading.

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3D Math Primer - Aimed at complete beginners to vector and matrix algebra.

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make object cube 1,100
for x=1 to 360
rotate object 1,x,x,0
next x

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