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Let’s imagine that the Earth is an ideal Sphere of unit Radius. Points A , B and C are some Cities on it: Moscow(A), New York(B) and Tokyo(C) for example. When you travel by the shortest way from Moscow to New York (AB) and then from New York to Tokyo(BC) also by the shortest way, the resulting geodesical arc on the Earth from Moscow to Tokyo (AC)being the shortest way from Moscow to Tokyo(AC)which IS BUT MULTIPLICATION product of Quaternions (AB) & (BC):
(AC)=(AB)@(BC),
with all needed requirements for multiplication @ being satisfied.
Every geodesical arc (AB)on the Unit Sphere correspondes to a Quaternion. If A=B then quaternion (ABB) is equal to 1. If A is diametrally opposite to B, then (AB) is equal to -1.The i j k quaternions being arcs along equator, Grinvich ang 90 degree after Grenvich meridians with longitude 90 degrees. Twisely taken i or j or k quternion gives rise to -1. etc.
There is nothing new about generalizing quaternions.
The most natural extension is to Cayley numbers,
where you lose associativity as well as commutativity, and
any further extension you are reduced to an algebra that is
only quasi-associative . . . This isn’t so nice.In other words–using the “obvious” generalization–as “n” increases,
the algebraic structure becomes necessarily cumbersome.
It is therefore not obvious that this becomes simpler than using
local charts on the smooth structure of S^n and just recognizing
that the geodesics on S^n are great circles of S^{n-1} to obtain
the new coordinates.Did you have a different idea in mind of a “generalized quaternion”?
S
For one, octonions are NOT even associative, so if you want an
associative algebra, the best you can do is quaternions.Octonions are a generalization, but you LOSE associativity.
You CAN generalize further, but you get nastier
power associative algebras like sedenions.All of this is ultimately cumbersome though.
However, if you are not using this machinery, and all you are doing is defining a “multiplication of arcs”, then in reality how is that any different from
standard vector addition? In reality, you are really describing a path product,
which is OK to define, but you won’t necessarily obtain a path you want . . .The composition of two distance-minimizing geodesics is
not necessarily a distance-minimizing geodesic!
Geodesics are only LOCALLY distance-minimizing.Example: consider two sequential geodesics on the same great circle–
if they are long enough, the composition creates a geodesic that is
not the length-minimizer you want.Steven
Yeah, but you are picking and choosing what rules you want.
After all, if you want commutativity, you are stuck with the complex numbers!See my other post.
S
Quaternions are not commutative. You lose commutativity when you
generalize to quaternions. You lose associativity when you generalize
to octonions. The higher order generalizations continue to get worse.S
Then your path AC is not well-defined!
If AB and BC both lie on the same geodesic, then going from
A to B to C forms a geodesic, but not necessarily length minimizing.In my example, you can have two geodesics going from A to C, one
is length minimizing and one is not. There is no obvious mechanism
that you can incorporate into the mathematics that will always
select the length minimizing one.S
>>When you move continually from B to C,
>>being at time t in the point X[t] on the arc BC,
>>the resulting product AB@BX[t] changes also continually from AB to AC.
>>This is a usual normal well-definition of the path AC.But this is exactly the problem. In my example, there are two legitimate
definitions of AC. If you want to force AC to be the shortest path, then
you will necessarily have an unaccounted-for discontinuity when the
“other AC” becomes the one you want. This creates an inconsistent definition.S
This is different and this is something you can do, but . . .
However, in this case, all you are really doing is giving
“different names” to the generators of SO(4), because you
are just using the SO(4) multiplication structure on these
new names.S
In my example of the two arcs on the same geodesic path, you can’t
force the continuity. If you are restricting to SHORTEST geodesic,
the path will instantaneously switch once the other geodesic option
becomes the shortest path.If you are making a definition in terms of arcs, then I think
you have to define it locally in a manifold sense.S
AC is not a length-minimizing geodesic anymore necessarily;
you get a geodesic, but not a length-minimizing one.This is the source of the well-defined problem.
If all you care about is geodesic preservation, then fine.
Products of the form AB@BC are defined in *that* case, but
then your comment that the products AB@BC represent
shortest paths on the n-sphere is not accurate.You have to decide what you want.
S
Yes, this is why I was making such a big deal about the
length-minimizing feature. You wanted to use “arc multiplication”
to generalize the notion of complex numbers and quaternions in a
new way.To be well-defined, I wanted to know what properties you wanted to
carry over into the generalization . . .S
With geodesic preservation only, there is no problem with
the AB@BC definition.Here’s an idea for your situation with AB@CD:
Consider parallel translations of CD to arcs BE . . .
You *may* have an issue with uniqueness of parallel translation,
depending on how the parallel translation is done.Recommendation: consider variations of CD to BE along arc BC.
Have fun,
S
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