Code

Java code to do conversion:

 public void set(Quat4d q1) {
   if (q1.w > 1) q1.normalise(); // if w>1 acos and sqrt will produce errors, this cant happen if quaternion is normalised
   angle = 2 * Math.acos(q1.w);
   double s = Math.sqrt(1-q1.w*q1.w); // assuming quaternion normalised then w is less than 1, so term always positive.
   if (s < 0.001) { // test to avoid divide by zero, s is always positive due to sqrt
     // if s close to zero then direction of axis not important
     x = q1.x; // if it is important that axis is normalised then replace with x=1; y=z=0;
     y = q1.y;
     z = q1.z;
   } else {
     x = q1.x / s; // normalise axis
     y = q1.y / s;
     z = q1.z / s;
   }
}

Derivation of Equations

from AxisAngle to Quaternion page we have:

1. --- q1.x = a1.x * Math.sin(a1.angle/2)
2. --- q1.y = a1.y * Math.sin(a1.angle/2)
3. --- q1.z = a1.z * Math.sin(a1.angle/2)
4. --- q1.w = Math.cos(a1.angle/2)

therefore from 4.

a1.angle/2 = Math.acos(q1.w)

5. --- a1.angle = 2 * Math.acos(q1.w)

and from 1.

a1.x = q1.x / Math.sin(a1.angle/2)

substituting from 5 gives:

a1.x = q1.x / Math.sin(Math.acos(q1.w))

from above diagram this gives:

a1.x = q1.x / Math.sqrt(1-q1.w*q1.w)

simarly for y and z:

a1.y = q1.y / Math.sqrt(1-q1.w*q1.w)

a1.z = q1.z / Math.sqrt(1-q1.w*q1.w)

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 quaternion for this rotation is: (0.7071+ i 0.7071)

So using the above result:

angle = 2 * acos(qw) = 2 * acos(0.7071) = 90 degrees
s = sqrt(1-qw*qw) = sqrt(0.5) = 0.7071
x = qx / s = 0.7071 / 0.7071 = 1
y = qy / s = 0
z = qz / s = 0

this gives the axis angle:

angle = 90 degrees
axis = 1,0,0

which agrees with the result shown here

Note:

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

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