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:
rightUp
reference orientation
heading = 0
attitude = 0
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0
which gives:
angle = 0
x = 0
y = 0
z = 0
Normalise axis, since angle is 0 axis can be anything so set axis to (1,0,0)
angle = 0 degrees
axis = 1,0,0
backUp
rotate by 90 degrees about y axis
heading = 90 degrees
attitude = 0
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0
which gives:
angle = 90 degrees
x = 0
y = 0.7071
z = 0
Normalising axis gives:
angle = 90 degrees
axis = 0,1,0
leftUp
rotate by 180 degrees about y axis
heading = 180 degrees
attitude = 0
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =1
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0
which gives:
angle = 180 degrees
x = 0
y = 1
z = 0
axis is already normalised
angle = 180 degrees
axis = 0,1,0
forwardUp
rotate by 270 degrees about y axis
heading = -90 degrees
attitude = 0
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =1
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0
which gives:
angle = 90 degrees
x = 0
y = -0.7071
z = 0
Normalising axis gives:
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 streight down orientations are known as singularities because they can be fully defined without using the bank value) post multiply above by 0.7071 + k 0.7071 to give:
upLeft
heading = 0
attitude = 90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =0.7071
s3 = sin(bank/2) =0
which gives:
angle = 90 degrees
x = 0
y = 0
z = 0.7071
Normalising axis gives:
angle = 90 degrees
axis = 0,0,1
heading = 90 degrees
attitude = 90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =0.7071
s3 = sin(bank/2) =0
which gives:
angle = 120 degrees
x = 0.5
y = 0.5
z = 0.5
Normalising axis gives:
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
upRight
heading = 180 degrees
attitude = 90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =0.7071
s3 = sin(bank/2) =0
which gives:
angle = 180 degrees
x = 0.7071
y = 0.7071
z = 0
axis is already normalised:
angle = 180 degrees
axis = 0.7071,0.7071,0
heading = -90 degrees
attitude = 90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =0.7071
s3 = sin(bank/2) =0
which gives:
angle = 120 degrees
x = -0.5
y = -0.5
z = 0.5
Normalising axis gives:
angle = 120 degrees
axis = -0.5774,-0.5774,0.5774
Or instead apply attitude -90 degrees (also a singularity): post multiply top row by 0.7071 - k 0.7071 to give:
downRight
heading = 0
attitude = -90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =-0.7071
s3 = sin(bank/2) =0
which gives:
angle = 90 degrees
x = 0
y = 0
z = -0.7071
Normalising axis gives:
angle = 90 degrees
axis = 0,0,-1
(equivilant rotation to:
angle = -90 degrees
axis = 0,0,1)
heading = 90 degrees
attitude = -90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =-0.7071
s3 = sin(bank/2) =0
which gives:
angle = 120 degrees
x = -0.5
y = 0.5
z = -0.5
Normalising axis gives:
angle = 120 degrees
axis = -0.5774,0.5774,-0.5774
downLeft
heading = 180 degrees
attitude = -90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =-0.7071
s3 = sin(bank/2) =0
which gives:
angle = 180 degrees
x = -0.7071
y = 0.7071
z = 0
axis is already normalised:
angle = 180 degrees
axis = -0.7071,0.7071,0
heading = -90 degrees
attitude = -90 degrees
bank = 0
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =0.7071
c3 = cos(bank/2) =1
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =-0.7071
s3 = sin(bank/2) =0
which gives:
angle = 120 degrees
x = 0.5
y = -0.5
z = -0.5
Normalising axis gives:
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: post multiply top row by 0.7071 + i 0.7071 to give:
heading = 0
attitude = 0
bank = 90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0.7071
which gives:
angle = 90 degrees
x = 0.7071
y = 0
z = 0
Normalising axis gives:
angle = 90 degrees
axis = 1,0,0
backRight
heading = 90 degrees
attitude = 0
bank = 90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0.7071
which gives:
angle = 120 degrees
x = 0.5
y = 0.5
z = -0.5
Normalising axis gives:
angle = 120 degrees
axis = 0.5774,0.5774,-0.5774
heading = 180 degrees
attitude = 0
bank = 90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0.7071
which gives:
angle = 180 degrees
x = 0
y = 0.7071
z = -0.7071
axis is already normalised:
angle = 180 degrees
axis = 0,0.7071,-0.7071
forwardLeft
heading = -90 degrees
attitude = 0
bank = 90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =0.7071
which gives:
angle = 120 degrees
x = 0.5
y = -0.5
z = 0.5
Normalising axis gives:
angle = 120 degrees
axis = 0.5774,-0.5774,0.5774
Apply bank +180 degrees: post multiply top row by i to give:
rightDown
heading = 0
attitude = 0
bank = 180 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =1
which gives:
angle = 180 degrees
x = 1
y = 0
z = 0
axis is already normalised:
angle = 180 degrees
axis = 1,0,0
backDown
heading = 90 degrees
attitude = 0
bank = 180 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =1
which gives:
angle = 180 degrees
x = 0.7071
y = 0
z = -0.7071
axis is already normalised:
angle = 180 degrees
axis = 0.7071,0,-0.7071
leftDown
heading = 180 degrees
attitude = 0
bank = 180 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =1
which gives:
angle = 180 degrees
x = 0
y = 0
z = 1
axis is already normalised:
angle = 180 degrees
axis = 0,0,1
forwardDown
heading = -90 degrees
attitude = 0
bank = 180 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =1
which gives:
angle = 180 degrees
x = 0.7071
y = 0
z = 0.7071
axis is already normalised:
angle = 180 degrees
axis = 0.7071,0,0.7071
Apply bank -90 degrees: post multiply top row by 0.7071 - i 0.7071 to give:
heading = 0
attitude = 0
bank = -90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =1
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =0
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =-0.7071
which gives:
angle = 90 degrees
x = -0.7071
y = 0
z = 0
Normalising axis gives:
angle = 90 degrees
axis = -1,0,0
(equivilant rotation to:
angle = -90 degrees
axis = 1,0,0)
backLeft
heading = 90 degrees
attitude = 0
bank = -90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =-0.7071
which gives:
angle = 120 degrees
x = -0.5
y = 0.5
z = 0.5
Normalising axis gives:
angle = 120 degrees
axis = -0.5774,0.5774,0.5774
heading = 180 degrees
attitude = 0
bank = -90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =1
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =-0.7071
which gives:
angle = 180 degrees
x = 0
y = 0.7071
z = 0.7071
axis is already normalised:
angle = 180 degrees
axis = 0,0.7071,0.7071
forwardRight
heading = -90 degrees
attitude = 0
bank = -90 degrees
angle = 2 * acos(c1c2c3 - s1s2s3)
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3 - s1 c2 s3
where:
c1 = cos(heading/2) =0.7071
c2 = cos(attitude/2) =1
c3 = cos(bank/2) =0.7071
s1 = sin(heading/2) =-0.7071
s2 = sin(attitude/2) =0
s3 = sin(bank/2) =-0.7071
which gives:
angle = 120 degrees
x = -0.5
y = -0.5
z = -0.5
Normalising axis gives:
angle = 120 degrees
axis = -0.5774,-0.5774,-0.5774
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