Quaternion to Angular Velocity

Quaternion to Angular Velocity – QuaternionExtensions.cs

using UnityEngine;

public static class QuaternionExtensions {

	//-----------------------------------------------------------------------------------------
	// Public Methods:
	//-----------------------------------------------------------------------------------------

	/// <summary>Returns a <c>Vector3</c> angular velocity representation of this <c>Quaternion</c>.</summary>
	public static Vector3 ToAngularVelocity(this Quaternion self) {

		// return a default vector if our magnitude overshoots.
		if (Mathf.Abs(self.w) > 1023.5f / 1024.0f) return new Vector3();

		// work out gain which will be our components' magnitude.
		float angle = Mathf.Acos(Mathf.Abs(self.w));
		float gain  = Mathf.Sign(self.w) * 2.0f * (angle / Mathf.Sin(angle));

		return new Vector3(self.x * gain, self.y * gain, self.z * gain);
	}
}

Quaternion to Angular Velocity – Vector3Extensions.cs

using UnityEngine;

public static class Vector3Extensions {

	//-----------------------------------------------------------------------------------------
	// Public Methods:
	//-----------------------------------------------------------------------------------------

	/// <summary>Returns a <c>Quaternion</c> from this angular velocity <c>Vector3</c>.</summary>
	public static Quaternion FromAngularVelocity(this Vector3 self) {

		// grab magnitude and return identity if there is none.
		float magnitude = self.magnitude;
		if (magnitude <= 0) return Quaternion.identity;

		// work out sin gain.
		float cos     = Mathf.Cos(magnitude * 0.5f);
		float sinGain = Mathf.Sin(magnitude * 0.5f) / magnitude;

		return new Quaternion(self.x * sinGain, self.y * sinGain, self.z * sinGain, cos);
	}
}