Contents  Introduction  Let's Get Started  Quaternions  Converting from  Quaternions  Multiplying  Quaternions  Conversion to  Quaternions  Demo  Conclusion  Printable version

Multiplying Quaternions

Since a unit quaternion represents an orientation in 3D space, the multiplication of two unit quaternions will result in another unit quaternion that represents the combined rotation. Amazing, but it's true.

Given two unit quaternions

Q1=(w1, x1, y1, z1);
Q2=(w2, x2, y2, z2);

A combined rotation of unit two quaternions is achieved by

Q1 * Q2 =( w1.w2 - v1.v2, w1.v2 + w2.v1 + v1*v2)

where

v1= (x1, y1, z1)
v2 = (x2, y2, z2)

and both . and * are the standard vector dot and cross product.

However an optimization can be made by rearranging the terms to produce

w=w1w2 - x1x2 - y1y2 - z1z2
x = w1x2 + x1w2 + y1z2 - z1y2
y = w1y2 + y1w2 + z1x2 - x1z2
z = w1z2 + z1w2 + x1y2 - y1x2

Of course, the resultant unit quaternion can be converted to other representations just like the two original unit quaternions. This is the real beauty of quaternions - the multiplication of two unit quaternions in 4D space solves gimbal lock because the unit quaternions lie on a sphere.

Be aware that the order of multiplication is important. Quaternion multiplication is not commutative, meaning

q1 * q2    does not equal    q2 * q1

Note: Both quaternions must refer to the same coordinate axis. I made the mistake of combining two quaternions from different coordinate axes, and I had a very hard time wondering why the result quaternion fails in certain angles only.

Next : Conversion to Quaternions