Maths - Axis-Angle to Euler - Sample Orientations

Sample Rotations

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:

angle = 0 degrees
axis = 1,0,0

heading = atan2(y * sin(angle)- x * z * (1 - cos(angle)) , 1 - (y2 + z2 ) * (1 - cos(angle)))
attitude = asin(x * y * (1 - cos(angle)) + z * sin(angle))
bank = atan2(x * sin(angle)-y * z * (1 - cos(angle)) , 1 - (x2 + z2) * (1 - cos(angle)))

sin(angle)=0
1-cos(angle)=0

heading = atan(0 / 1)
attitude = asin(0)
bank = atan(0 / 1)

heading = 0
attitude = 0
bank = 0

backup

rotate by 90 degrees about y axis:

angle = 90 degrees
axis = 0,1,0

heading = atan2(y * sin(angle)- x * z * (1 - cos(angle)) , 1 - (y2 + z2 ) * (1 - cos(angle)))
attitude = asin(x * y * (1 - cos(angle)) + z * sin(angle))
bank = atan2(x * sin(angle)-y * z * (1 - cos(angle)) , 1 - (x2 + z2) * (1 - cos(angle)))

sin(angle)=1
1-cos(angle)=1

heading = atan2(1 , 0)
attitude = asin(0)
bank = atan2(0 , 1)

heading = 90 degrees
attitude = 0
bank = 0

leftup

rotate by 180 degrees about y axis:

angle = 180 degrees
axis = 0,1,0

heading = atan2(y * sin(angle)- x * z * (1 - cos(angle)) , 1 - (y2 + z2 ) * (1 - cos(angle)))
attitude = asin(x * y * (1 - cos(angle)) + z * sin(angle))
bank = atan2(x * sin(angle)-y * z * (1 - cos(angle)) , 1 - (x2 + z2) * (1 - cos(angle)))

sin(angle)=0
1-cos(angle)=2

heading = atan2(0 , -1)
attitude = asin(0)
bank = atan2(0 , 1)

heading = 180 degrees
attitude = 0
bank = 0

forwardup

rotate by 270 degrees about y axis:

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

heading = atan2(y * sin(angle)- x * z * (1 - cos(angle)) , 1 - (y2 + z2 ) * (1 - cos(angle)))
attitude = asin(x * y * (1 - cos(angle)) + z * sin(angle))
bank = atan2(x * sin(angle)-y * z * (1 - cos(angle)) , 1 - (x2 + z2) * (1 - cos(angle)))

sin(angle)=-1
1-cos(angle)=1

heading = atan2(-1 ,0)
attitude = asin(0)
bank = atan2(0 , 1)

heading = -90 degrees
attitude = 0
bank = 0
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