Maths - Axis Angle to Quaternion Examples 90 degree steps

Maths - Axis-Angle to Quaternion - 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

angle = 0 degrees
axis = 1,0,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 1

rotate by 90 degrees about y axis

angle = 90 degrees
axis = 0,1,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 + j 0.7071

rotate by 180 degrees about y axis

angle = 180 degrees
axis = 0,1,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = j

rotate by 270 degrees about y axis

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

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 - j 0.7071

(equivilant rotation to:
-0.7071 + j 0.7071)

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)

angle = 90 degrees
axis = 0,0,1

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 + k 0.7071

upForward
angle = 120 degrees
axis = 0.5774,0.5774,0.5774

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

angle = 180 degrees
axis = 0.7071,0.7071,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = i 0.7071 +j 0.7071

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

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

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

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 - k 0.7071

(equivilant rotation to:
-0.7071 + k 0.7071)

downBack

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = -i 0.7071 + j 0.7071

(equivilant rotation to:
i 0.7071 - j 0.7071)

downForward

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

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

rightForward

angle = 90 degrees
axis = 1,0,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 + i 0.7071

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

leftBack

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = j 0.7071 - k 0.7071

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

Apply bank +180 degrees:

angle = 180 degrees
axis = 1,0,0

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = i

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = i 0.7071 - k 0.7071

angle = 180 degrees
axis = 0,0,1

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = k

angle = 180 degrees
axis = 0.7071,0,0.7071

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = i 0.7071 + k 0.7071

Apply bank -90 degrees:

rightBack

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

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = 0.7071 - i 0.7071

(equivilant rotation to:
-0.7071 + i 0.7071)

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

leftForward

angle = 180 degrees
axis = 0,0.7071,0.7071

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

q = j 0.7071 + k 0.7071

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

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

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

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Visualizing Quaternions by Andrew J. Hanson

Other Math Books

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

Dark Basic Professional Edition - It is better to get this professional edition

This is a version of basic designed for building games, for example to rotate a cube you might do the following:
make object cube 1,100
for x=1 to 360
rotate object 1,x,x,0
next x

Game Programming with Darkbasic - book for above software

Discussion about using Darkbasic for physics simulation

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progam

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http://sourceforge.net/projects/mjbworld/

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