What is the rationale behind this method of computing the w-component from MD5 mesh files?

Question

In tutorials regarding the MD5Mesh format, like this one, we often see code like

float t = 1.0f - (q.x * q.x) - (q.y * q.y) - (q.z * q.z);

if (t < 0.0f) {
   q.w = 0.0f;
} else {
   q.w = -sqrt (t);
}

for computing the w-component of a unit quaternion given x, y, z. Why do we obtain w in this manner?

Answer 1

The MD5 file format assumes unit quaternions, which are of norm one. The norm of some quaternion q is:

|q| = sqrt((x * x) + (y * y) + (z * z) + (w * w))

You can rearrange this equation in terms of w by squaring both sides and then subtracting the terms containing x, y and z to yield:

(w*w) = (|q|*|q|) - (x*x) - (y*y) - (z*z)

Since the quaternion is assumed to be of norm 1, the |q|*|q| term is thus 1, and that's how you arrive at the initialization of t in your code snippet. Taking the square root of t yields the value of the final component, the sign of which is chosen as an orientation convention.