Understanding the derivative of a motor

Projective GA
1

Hello bivector forum!

I’m an enthusiastic beginner when it comes to PGA and I have a basic comprehension question about the ‘May The Forque Be With You’ guide (May the Forque Be with You!):

I just don’t understand what is meant by the derivative of the motor. If the motor represents the current position and orientation, wouldn’t the derivative of the motor be a velocity? If so, what would be the difference between the bivector representing the velocity and the derivative of the motor? I’m just trying to somehow link this concept to my basic understanding of position/velocity, if that’s even possible.

I would appreciate if someone here could explain this to me (in simple terms would be nice). Thank you!

2

Hi Roberti,

Welcome to the forum - and to (P)GA!

The derivative of position is (linear) velocity, the derivative of orientation is (angular) velocity, and similarly, the derivative of a motor gives the ‘velocity’ on the 6-dimensional motor manifold.

So, given a motor M, you can find the velocity it gives to a point p by considering the derivative not of M but of Mp\tilde M.

It is also instructive to consider what the derivative of a motor then is.To do this, we remember that each motor that represents a valid rigid body motion also has a norm of 1:

M\tilde M = 1

As the motor evolves, this condition must continue to hold true, so in other words, the derivative of this expression must be zero. Hence using a dot to denote the time derivative we can write

\dot{M \tilde M} = 0

Applying the product rule for derivatives we find

\dot M\tilde M + M \dot{\tilde M} = 0

and rearranging

\dot M\tilde M = - M \dot{\tilde M}

however, using that the reverse and derivative commute, we can use \widetilde{M \dot{\tilde M}} = \dot M \tilde M and write

\dot M\tilde M = - \widetilde{\dot M \tilde M}

This tells us that the quantity \dot M \tilde M changes sign under reversal. But in 3DPGA, we know bivectors are the only even elements that do that, so \dot M \tilde M, or the derivative of M at the origin, also known as the logarithm of M, must be a bivector.

\ln M = \dot M \tilde M = b

Hope that helps a bit,

S.

3

Hi Enki,

Thanks for the explanation! Although I don’t quite understand it yet (mainly because I still lack some basics in geometric algebra), it definitely helps me. I will continue to study and try to understand it more.

Thank you so much!

Robert