Quaternion Question

[Johnny5]

I'm not sure the best category to post this in, so I'm putting it here.

 

In the the link I am about to post, there is a 3D game programmer Diana Gruber discussing quaternions. During the thread, you can see her progressing towards understanding them. In trying to figure them out, she was led all the way to their inventor Gibbs. I don't know where she finally found Gibbs' work, but she did, and she pasted his equations.

 

Gibbs equations for quaternions

 

My question is about the matrix which Gibbs developed.

 

Is anyone here comfortable enough with quaternions to discuss them?

 

I want to understand that matrix.

 

PS:

 

A few posts later, Ms. Gruber writes:

 

> Then what the $%^?@! are we arguing about?

 

Imaginary numbers. Don't need 'em. Gibbs got all the way there without ever having to take the square root of -1.

 

Diana

[matt grime]

Yes, but Hamilton is usually credited with creating the Quarternions. I've no idea who Gibbs is.

[Johnny5]

Yes, but Hamilton is usually credited with creating the Quarternions. I've no idea who Gibbs is.

 

I presume the same Gibbs that Gibbs free energy is named after.

 

Diana posted that stuff as Gibbs work, although elsewhere I saw Hamilton credited with it, but I don't buy that "etched it in a bridge" story. She was trying to understand quaternions, which is what led her to that Matrix. And that matrix has something to do with rotations. And apparently, something about quaternions avoids gimbal lock, which afflicts Euler angle approach to rotations.

 

Regards

[matt grime]

Ok, don't buy the bridge thing (I don't think many people do), but Hamilton is still widely credited with inventing the quarternions.

 

Are you going to post "that matrix"? I looked at the page (light font dark background) and didn't see anything like that.

[Johnny5]

 

Are you going to post "that matrix"? I looked at the page (light font dark background) and didn't see anything like that.

 

Sure' date=' its on the link but here it is:

 

F =

 

(1+a²+b²+c²)^-1 *

 

[(1+a²-b²-c²) (2ab-2c) (2ac+2b)]

[(2ab+2c) (1-a²+b²-c²) (2bc-2a)]

[(2ac-2b) (2bc+2a) (1-a²-b²+c²)]

[MetaFrizzics]

Nowadays we are taught vectors in school as though they were the original or main thing, almost a trivial 'obvious' solution to the problem of combining forces.

 

Historically, however, physics was branching in two different directions just after Newton:

(1) toward the (Newtonian) deterministic vector-like approach, in which one predicts the evolution of a system by its current state, (velocities & positions of particles), and another more esoteric idea

(2) the 'least action' /work function (Leibnitzian) concept, in which difficult problems could be solved by bypassing detailed individual knowledge of components, but instead concentrate on the evolution of a whole system.

 

Regardless of exact credits (following Euler and Gauss), in England Hamilton and in Germany Grassmann were independantly (? some suspicion there) working on the 2nd branch of physics, which was to evolve into Hamiltonian or Analytical Variational Mechanics, or Calculus of Variations etc. In this process, Hamilton erected the most complete 4-dimensional system (Quaternions) and put it to use solving difficult physical problems. (by the way the DEL operator was first formed by Hamilton).

 

It took Tait and Maxwell however to explore the fantastic territory of Quaternions and create the first real 'field theory' Classical Electromagnetic Theory. Maxwell based his theory on Hamilton's system, although pointed the way to breaking it down into simpler and more practical parts and applications. (Maxwell rejected Grassman's version, and Weber's Electromagnetics, in his own approach although acknowledging their value).

 

But this was still impenetrable until (Gibbs and mainly) Heaviside followed Maxwell's hints and virtually singlehandedly invented both modern 'vectors' and Operational Calculus. So what? Well, Maxwell's original TWELVE messy equations and confusing application of Quaternions was completely cleaned up by Heaviside and modern vectors and Electrical Engineering was born. It was Heaviside who invented and coined virtually all the modern terms like resistivity permeability etc., and who unfortunately goes largely unrecognized for his massive contribution to the age of electricity and electronics. Heaviside systematized Maxwell and created the now common FOUR Maxwell Equations in the form they are now known, and made the science of electrical engineering possible.

 

Just as we now learn Newton in vector form, so we also learn electronics in vector form, thanks to Heaviside's vector calculus.