Correct way to integrate scaled axis angular velocity into a quaternion

Question

Across my travels on the internet I have came across two different ways of integrating a scaled axis angular velocity into a quaternion.

The first way converts the angular velocity into an axis angle quaternion like so

Quaternion q = new Quaternion(vector.normalised(), vector.getLength());

Where the constructor looks like:

float sinHalfAngle = (float) Math.sin(angle / 2);
float cosHalfAngle = (float) Math.cos(angle / 2);

this.x = axis.x * sinHalfAngle;
this.y = axis.y * sinHalfAngle;
this.z = axis.z * sinHalfAngle;
this.w = cosHalfAngle;

And then multiplies the existing quaternion by that new quaternion

this.rotateSelf(q);

The second approach also converts the velocity into a quaternion, but uses a different method:

Quaternion change = Quaternion.mul(
    new Quaternion(x * 0.5f, y * 0.5f, z * 0.5f, 0),
    this
);

It then adds the quaternions together, scaled by delta T:

this.w += change.w * scale;
this.x += change.x * scale;
this.y += change.y * scale;
this.z += change.z * scale;

Both approaches produce a rotation, however the first approach rotates the object much faster than the second. I was wondering which is the 'more' correct way, or if they are just different methods used for different units perhaps? (ie: degrees vs radians?)

Thanks very much if you can help.

Answer 1

When Microsoft created the XNA framework (I know you're using Java, but the math is the same), they created a built in method to create a Quaternion from an axis and angle. The method they used (and you can "reflect" their managed code to verify it, see below) is the same as your first snippet:

public static Quaternion CreateFromAxisAngle(Vector3 axis, float angle)
{
    Quaternion quaternion;
    float num2 = angle * 0.5f;
    float num = (float) Math.Sin((double) num2);
    float num3 = (float) Math.Cos((double) num2);
    quaternion.X = axis.X * num;
    quaternion.Y = axis.Y * num;
    quaternion.Z = axis.Z * num;
    quaternion.W = num3;
    return quaternion;
}