A simple problem, with your arm out t-pose palm flat (facing down) do this:

1. Rotate you arm down to the side, then forward (You’ll notice the bicep faces up)
2. Now go back to your arm out. Rotate it forward (the bicep will face to the right or left now)

We have an oddity here, and I think it’s one of the founding principles of biomechanical rotation, it ‘resolves’ itself. When two rotations meet the union causes another plane of freedom to be introduced – i.e the top of the bicep will twist 45 degrees with your arm going forward and -45 going backwards. Looking at this from a math perspective its spherical rotation. (a quaternion)This spherical rotation similar to a quaternion is what stops the arm from twisting itself off its joint. The muscles are are treating its ball and socket joint as a spherical rotation or in other words a quaternion. Now this may not seem interesting but this is before there’s been any rotation of the elbow.I.e the twist of the upper arms is brought on by the constant rotation of the shoulder  and its resolution or twist,  is brought on by the rotation of the elbow. This is why it can be hard to get a frame of reference for the twist. I’ll see if i can update this with some pics.

1. November 13, 2007

Interesting topic Charles! I have been thinking on that same problem many times now. I`m not good in mathematics as you are /oh boy I hated it at school :)/ and It`s nice someone to interpret the real world things to mathematic point of view.

2. Aline Cren
November 16, 2007

I think charles’ description is good, but not entirely correct.

3. November 16, 2007

Thanks Martin – basically what I tend to do is look at the real world and nature. I class it as 10% natural inspiration, 70% turning it into math and 20% making it friendly for the animator.

I basically took my arm and watched it rotate – i did the same thing with a pen and found out something odd was going on.

Essentially we naturally resolve upper arm twist, because if our muscles were bound rigidly we’d twist our arms off!

What happens when our arm goes foward is that the arm twists about two axis – and in doing so lies along a new axis – we can say this because it is exactly the opposite when the arm goes backwards.

This natural resolution of the arm, is essentially what a quaternion does. This new axis could be consider the equator line of a 4D hemisphere. We know this because quats can still flip once they go past this line, and appear on the other side, unless they run along the line, slerping between multiple quats. This is known as ‘duriac’ or ‘the belt trick’