View Full Version : The inverse of a quaternion.

lgc_ustc

01-24-2004, 11:18 AM

Hello, all,

I am working on the QuakeIII md3 model loading code, everything works fine now except that the rotation interpolating part. This is done using crazy quaternions, and the problems is that the torso part rotates in the reverse direction of what it should do. I try to inverse the quaternion so that I can reverse the direction the torso rotates. Can anybody show me how to? Thanks.

orbano

01-24-2004, 12:51 PM

http://skal.planet-d.net/demo/matrixfaq.htm

[This message has been edited by orbano (edited 01-24-2004).]

m i s s i l e

01-26-2004, 12:54 PM

The inverse of a quaternion q.

Lets assume q is normalized, i.e. |q| = 1.

Usually, normalized quaternions are associated,

with rotations in R³. Rotations in R³ are

orthogonal matrices having determinant +-1.

The definition of an orthogonal matrix says

that the inverse equals its transpose.

So, one "solution" (for people having only a minor

knowledge of quaternions) is to convert the quat.

to a rotation in R³, transpose that rotation

and convert it back to a quaternion. This will give

you the inverse quaternion.

But this is not necessary, its far more easier.

If the quat. q is normalized, then the inverse

quat. q^(-1) of q is given by :

q^(-1) = q^t

q^t is the so-called conjugate (which can be

viewed as of transposing the equivalent rotation

matrix). Conjugation (in a complexified space) is

done by inverting the signs of the imaginary

numbers. For example, the conjugate of a complex

numbers z = a + ib is z^t = a - ib. And this is the

same with quaternions:

Let q = a + ib + jc + kd be a quaternion, then

q^t = a - ib - jc - kd

is the conjugate quaternion of q.

So the inverse is easily be found.

A side: Being the inverse of an element implies

that, if you combine the inverse of an element

with the element itself then you will get the

identity element.

For a normalized quat this means:

q * q^t must equal 1!

cu,

m i s s i l e

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