Unit quaternions

Quaternions can be normalized just like vectors. Normalized quaternions represent only a rotation and non-normalized quaternions introduce a skew. In the context of game animation, quaternions should be normalized to avoid adding a skew to the transform.

To normalize a quaternion, divide each component of the quaternion by its length. The resulting quaternion's length will be 1. This can be implemented as follows:

1. Implement the normalize function in quat.cpp and declare it in quat.h:

   void normalize(quat& q) {

      float lenSq = q.x*q.x + q.y*q.y + q.z*q.z + q.w*q.w;

      if (lenSq < QUAT_EPSILON) {

         return;

      }

      float i_len = 1.0f / sqrtf(lenSq);

      q.x *= i_len;

      q.y *= i_len;

      q.z *= i_len;

      q.w *= i_len;

   }

2. Implement the normalized function in quat.cpp, and declare it in quat.h:

   quat normalized(const quat& q) {

      float lenSq = q.x*q.x + q.y*q.y + q.z*q.z + q.w*q.w;

      if (lenSq < QUAT_EPSILON) {

         return quat();

      }

      float il = 1.0f / sqrtf(lenSq); // il: inverse length

      return quat(q.x * il, q.y * il, q.z * il,q.w * il);

   }

There is a fast way of inverting any unit quaternion. In the next section, you will learn how to find the conjugate and inverse of a quaternion and their relationship when it comes to unit quaternions.