Example of quaternion on quaternion multiplication using the C++ Eigen library.

quaternionMultiplicationExample.cpp

Eigen::Quaterniond MyWorld::quatMult(Eigen::Quaterniond q1, Eigen::Quaterniond q2) {
    Eigen::Quaterniond resultQ;
    resultQ.setIdentity();

    resultQ.w() = q1.w() * q2.w() - q1.vec().dot(q2.vec());
    resultQ.vec() = q1.w() * q2.vec() + q2.w() * q1.vec() + q1.vec().cross(q2.vec());

    return resultQ;
}

Are you sure? A coworker added some code that is nearly identical to yours. (Including the initialization of resultQ to identity. Which is probably redundant since you overwrite all of its state.) On your blog you say “Unfortunately, it looks like the standard * operator performs normal multiplication, not the special quaternion multiplication required by an actual quaternion.” I am not sure what that means, but when I tried an example, they appear to compute the same value:

Eigen::Quaterniond a(1, 2, 3, 4);
Eigen::Quaterniond b(5, 6, 7, 8);
a.normalize();
b.normalize();
Eigen::Quaterniond c = a * b;
Eigen::Quaterniond d = quatMult(a, b);

produces:

a: (w=0.182574, x=0.365148, y=0.547723, z=0.730297)
b: (w=0.379049, x=0.454859, y=0.530669, z=0.606478)
c: (w=-0.830455, x=0.166091, y=0.415227, z=0.332182)
d: (w=-0.830455, x=0.166091, y=0.415227, z=0.332182)

Not a rigorous proof, but at least brings into question your original claim about Eigen’s operator* for quaternions.