Prerequisites
Definition of terms:
Equations
angle = 2 * acos(c_{1}c_{2}c_{3}  s_{1}s_{2}s_{3})
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3  s1 c2 s3
to normalise divide x,y and z by:
sqrt(x^{2} + y^{2} + z^{2}) = sqrt((s1 s2 c3 +c1 c2
s3)^{2}+(s1 c2 c3 + c1 s2 s3)^{2}+(c1 s2 c3  s1 c2 s3)^{2)}
where:
 c_{1} = cos(heading / 2)
 c_{2} = cos(attitude / 2)
 c_{3} = cos(bank / 2)
 s_{1} = sin(heading / 2)
 s_{2} = sin(attitude / 2)
 s_{3} = sin(bank / 2)
Code
public final void rotate(double heading, double attitude, double bank) {
// Assuming the angles are in radians.
double c1 = Math.cos(heading/2);
double s1 = Math.sin(heading/2);
double c2 = Math.cos(attitude/2);
double s2 = Math.sin(attitude/2);
double c3 = Math.cos(bank/2);
double s3 = Math.sin(bank/2);
double c1c2 = c1*c2;
double s1s2 = s1*s2;
w =c1c2*c3  s1s2*s3;
x =c1c2*s3 + s1s2*c3;
y =s1*c2*c3 + c1*s2*s3;
z =c1*s2*c3  s1*c2*s3;
angle = 2 * Math.acos(w);
double norm = x*x+y*y+z*z;
if (norm < 0.001) { // when all euler angles are zero angle =0 so
// we can set axis to anything to avoid divide by zero
x=1;
y=z=0;
} else {
norm = Math.sqrt(norm);
x /= norm;
y /= norm;
z /= norm;
}
}
Derivation of Equations
Start from quaternion to axis angle as shown here:
angle = 2 * acos(qw)
x = qx / sqrt(1qw*qw)
y = qy / sqrt(1qw*qw)
z = qz / sqrt(1qw*qw)
Substitute from Euler to Quaternion as shown here:
qw = c1 c2 c3  s1 s2 s3
qx = s1 s2 c3 +c1 c2 s3
qy = s1 c2 c3 + c1 s2 s3
qz = c1 s2 c3  s1 c2 s3
where:
 c_{1} = cos(heading / 2)
 c_{2} = cos(attitude / 2)
 c_{3} = cos(bank / 2)
 s_{1} = sin(heading / 2)
 s_{2} = sin(attitude / 2)
 s_{3} = sin(bank / 2)
So, removing common factors which means that x,y,z is no longer normalised
angle = 2 * acos(c_{1}c_{2}c_{3}  s_{1}s_{2}s_{3})
x = s1 s2 c3 +c1 c2 s3
y = s1 c2 c3 + c1 s2 s3
z = c1 s2 c3  s1 c2 s3
to normalise divide x,y and z by:
x^{2} + y^{2} + z^{2} = (s1 s2 c3 +c1 c2 s3)^{2}+(s1
c2 c3 + c1 s2 s3)^{2}+(c1 s2 c3  s1 c2 s3)^{2}
Issues
If you have a different result from that shown on this page it may be that
you are using different standards, I have tried to keep the standards consistant
accross this site and I have tried to define the standards that I am using here.
One of these standards is that the order that the euler rotations are applied
is 'NASA standard aeroplane' it uses a slightly different coordinate definition
from VRML (z and y axis swapped). Also when working with aeroplanes we often
work in terms of the position of external objects relative to the aircraft (i.e.
the inverse of its position transform as explained
here). therefore to get the expression normally used with NASA aeroplane
we invert all inputs (change sign of every term with odd number of sine terms)
invert output (conjugate quaternion) swap z and y for inputs (swap z and y columns)
swap z and y for outputs (swap z and y rows). In the case of aircraft it can
make sense to work in terms of the local frame of reference of the aircraft
looking out, I have a version of this page that does that here,
but be careful as this will no longer be compatible with the rest of this site.
However if you have checked these things and you still have a discrepency then
I have probably made an error so please let me know
here.
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
 c_{1} = cos(heading / 2) = 1
 c_{2} = cos(attitude / 2) = 1
 c_{3} = cos(bank / 2) = 0.7071
 s_{1} = sin(heading / 2) = 0
 s_{2} = sin(attitude / 2) = 0
 s_{3} = sin(bank / 2) = 0.7071
angle = 2 * acos(c_{1}c_{2}c_{3} + s_{1}s_{2}s_{3})
= 2 * acos(0.7071) = 90 degrees
x = c_{1}c_{2}s_{3}  s_{1}s_{2}c_{3}
= 0.7071
y = c_{1}s_{2}c_{3} + s_{1}c_{2}s_{3}
= 0
z = s_{1}c_{2}c_{3}  c_{1}s_{2}s_{3}
= 0
So the axis is: (0.7071,0,0)
This can be normalised to: (1,0,0)
So this gives the correct result.
Angle Calculator and Further examples
I have put a java applet here which allows the values to be entered and the converted values shown along with a graphical representation of the orientation.
Also further examples in 90 degree steps here
This site may have errors. Don't use for critical systems.