Definition of terms:

• Euler Angles
• Axis Angle

Start from quaternion to axis angle as shown here:

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

Substitute from Euler to Quaternion as shown here:

qw = c₁c₂c₃ + s₁s₂s₃
qx = c₁c₂s₃ - s₁s₂c₃
qy = c₁s₂c₃ + s₁c₂s₃
qz = s₁c₂c₃ - c₁s₂s₃

where:

• c₁ = cos(heading / 2)
• c₂ = cos(attitude / 2)
• c₃ = cos(bank / 2)
• s₁ = sin(heading / 2)
• s₂ = sin(attitude / 2)
• s₃ = sin(bank / 2)

So, removing common factors which means that x,y,z is no longer normalised

angle = 2 * acos(c₁c₂c₃ + s₁s₂s₃)
x = c₁c₂s₃ - s₁s₂c₃
y = c₁s₂c₃ + s₁c₂s₃
z = s₁c₂c₃ - c₁s₂s₃

to normalise divide x,y and z by:

x² + y² + z² = (c₁c₂s₃ - s₁s₂c₃)²+(c₁s₂c₃ + s₁c₂s₃)²+(s₁c₂c₃ - c₁s₂s₃)²

Issues

• For issues concerning acos see here.
• Most trig functions (except openGL) use radians not degrees.

Example

we take the 90 degree rotation from this:

to this:

As shown here the axis angle for this rotation is:

heading = 0 degrees
bank = 90 degrees
attitude = 0 degrees

• 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

angle = 2 * acos(c₁c₂c₃ + s₁s₂s₃) = 2 * acos(0.7071) = 90 degrees
x = c₁c₂s₃ - s₁s₂c₃ = 0.7071
y = c₁s₂c₃ + s₁c₂s₃ = 0
z = s₁c₂c₃ - c₁s₂s₃ = 0

So the axis is: (0.7071,0,0)

This can be normalised to: (1,0,0)

So this gives the correct result, see here, but we have to be very careful about the following issues:

• Most maths libraries use radians instead of degrees (apart from OpenGL).