Maths - Quaternion to Axis Angle examples 90 degree steps

Maths - Quaternion to Axis-Angle - Sample Orientations

Sample Rotations

In order to try to explain things I thought it might help to work out a simple case where rotations are only allowed in mutiples 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 axies in 90 degree multiples there are 24 possible orientations as in the table shown below.

In the following table we will need to know what quadrant the results are in, so I have taken some sample results from Math.atan2

heading applied first giving 4 possible orientations:

reference orientation

q = 1

sqrt(1-qw*qw) = 0

to avoid divide by zero just set axis to arbitary value (does not matter as angle is zero):

angle =0
x = 1
y = 0
z = 0

angle = 0 degrees
axis = 1,0,0

rotate by 90 degrees about y axis

q = 0.7071 + j 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,1,0

rotate by 180 degrees about y axis

q = j

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,1,0

rotate by 270 degrees about y axis

q = 0.7071 - j 0.7071

(equivilant rotation to:
-0.7071 + j 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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 streight down orientations are known as singularities because they can be fully defined without using the bank value)

q = 0.7071 + k 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,0,1

upForward

q = 0.5 + i 0.5 + j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = 0.5774,0.5774,0.5774

q = i 0.7071 +j 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0.7071,0.7071,0

upBack

q = 0.5 - i 0.5 - j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

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

q = 0.7071 - k 0.7071

(equivilant rotation to:
-0.7071 + k 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

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

downBack

q = 0.5 - i 0.5 + j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

q = -i 0.7071 + j 0.7071

(equivilant rotation to:
i 0.7071 - j 0.7071)

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

downForward

q = 0.5 + i 0.5 - j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

Normally we dont go beond attitude + or - 90 degrees because thes are singularities, instead apply bank +90 degrees:

rightForward

q = 0.7071 + i 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 1,0,0

q = 0.5 + i 0.5 + j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

leftBack

q = j 0.7071 - k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

q = 0.5 + i 0.5 - j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

Apply bank +180 degrees:

q = i

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 1,0,0

q = i 0.7071 - k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

q = k

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,0,1

q = i 0.7071 + k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0.7071,0,0.7071

Apply bank -90 degrees:

rightBack

q = 0.7071 - i 0.7071

(equivilant rotation to:
-0.7071 + i 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

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

q = 0.5 - i 0.5 + j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

leftForward

q = j 0.7071 + k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,0.7071,0.7071

q = 0.5 - i 0.5 - j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

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

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see also:

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Book Shop - Further reading.

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Visualizing Quaternions by Andrew J. Hanson

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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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