Exploring Unit Quaternions and Their Relationship to Spheres

In summary, the conversation discusses the use of unit quaternions in identifying the 3-sphere and their relationship with the even-grade subalgebra. It is noted that quaternions can rotate both vectors and planes, and there is a one-to-one correspondence between planes and their normal vectors. The conversation also touches on defining a distance between unit quaternions and the use of the 3-sphere and Euclidean space. The formula for calculating the geodesic distance between two unit quaternions is presented, as well as a comparison between chord-length and arc-length as distances. The conversation concludes with a question about the terminology for two distances that satisfy a certain property.
  • #1
mnb96
715
5
Hello,
I read somewhere that the set of unit quaternions identifies the [tex]\mathcal{S}^3[/tex] sphere.
This makes sense; however, what happens if we consider instead a quaternion as an element of the even-grade subalgebra [tex]\mathcal{C}\ell^+_{3,0}[/tex] ?

Now a unit quaternion is represented as a scalar-plus-bivector [tex]p+\mathbf{B}q[/tex] which can be written in the form [tex]cos(\alpha)+\mathbf{B}sin(\alpha)[/tex] where [itex]\alpha[/itex] is an angle on the plane B.

So why can´t we consider a quaternion as an element of [tex]\mathcal{S}^2[/tex] instead?
 
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  • #2
This is exactly how it is nowadays. This way quaternions can rotate both vectors and planes (or axial vectors) by just one simple formula. In other words: quaternions are Hodge dual to vectors:
[tex]\mathbf{B}=i\mathbf{b}=\star \mathbf{b}[/tex], where [tex]i=e_1e_2e_3[/tex] is in the center of the algebra with [tex]i^2=-1[/tex].
 
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  • #3
Thanks arkajad!
if I understood correctly, here the interesting observation is that in 3 dimensional space (only!) the dual of a vector is indeed a bi-vector (and vice-versa) => there is a one-to-one correspondence between planes and their normal-vectors.

However, let's say that now, we want to define a distance between unit-quaternions.
In the case of unit complex numbers this is easy, because we know that each unit complex number identifies one point on the [itex]\mathcal{S}^1[/itex] sphere: the unit circle.

When dealing with unit-quaternions I guess we are forced to consider them as elements of the [itex]\mathcal{S}^3[/itex] sphere and use (for example) the shortest-arc on the 3-sphere as distance measure. This is because there is not necessarily a one-to-one corrspondence between unit-quaternions and points on the [itex]\mathcal{S}^2[/itex] sphere.

Is this correct?
 
  • #4
Well, you can take the distance from the 4-dimensional Euclidean space.
 
  • #5
thanks arkajad!
so, summarizing: the equivalent of the unit-circle for complex numbers is the 3-sphere for unit-quaternions.

It is true that we can indeed use the euclidean distance.
I wouldn't want to go too much off-topic, but is there a "mathematical definition" to express the relationship between the shortest arc on the sphere, and the euclidean distance between point on the sphere?
 
  • #6
I wrote originally:

You can look at it in a different way. Each unit quaternion is just four real numbers with squares adding to 1. So, you can associate with it a quantum vector state of a qubit. If |q>,|q'> are such states then, assuming q and q' are not too far from each other, the geodesic distance D(q,q') is given by the formula

[tex]\cos^2(D(q,q'))=|<q|q'>|^2[/tex]

To relate it to physics - see, for instance, "http://arxiv.org/abs/quant-ph/0509017" [Broken]", Eq. (13).


But no, that was wrong!

From my calculations the geodesic distance between two unit quaternions q,q' is given by:

[tex]D(q,q')=\arccos (|1-\frac12||q-q'||^2|)[/tex]

See the attached extract from Hanson, "Visualizing quaternions".
 

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  • #7
Uhm...
I am bit confused now.

The formula that is shown on the scanned page is different than yours: Hanson basically computes the scalar product between two (unit?) quaternions and says that it is the cosine of the angle between them. He uses the ordinary "euclidean inner product".

You instead seem to use the inner-product commonly associated with the conformal model of geometric algebra [itex]\mathcal{C}\ell_{4,1}[/itex], which is: [tex]-\frac12(q-p)^2[/tex]. Then you say that the cosine of the angle is "1 minus the inner product".

What is the difference/advantages/disadvantages between the two?
Personally, I was even thinking of using the bare Euclidean distance between points on the 3-sphere. it works for both the unit circle in 2D and the unit sphere in the 3D.
 
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  • #8
Simple calculation in the 4d Euclidean space :

[tex]||q-p||^2=(q-p,q-p)=||q||^2+||p||^2-2(p,q)=2(1-(p,q))[/tex]

You can calculate |(p,q)| from this.
 
  • #9
Ops...Sorry!
I should have not missed that!
I am evidently tired at this time in the evening :)
You compute the d(q,q') which is a chord-length and then simply retrieve the arc-length.
Now everything is clear.
Thanks a lot!


One very last thing: when we consider the two aforementioned distances, chord-length and arc-length (respectively [itex]d_c[/itex] and [itex]d_a[/itex]), are these distances "said to be something"?
In other words, is there a definition in the literature to denote two distances which have the property [itex]d_c(x,y)\leq d_c(x,z) \Leftrightarrow d_a (x,y)\leq d_a (x,z)[/itex] for all x,y,z

Thanks again, you fully answered my original question.
 
  • #10
Well, look at the graph of the function [tex]f(d)=\arccos(1-d^2/2),\quad 0<d<1[/tex]

You see that [tex]d_1<d_2[/tex] iff [tex]f(d_1)<f(d_2)[/tex] - the derivative being positive!. So at least there you have what you need.
 

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  • #11
Yes, my question was essentially:
- is there a name in literature to call two distances that satisfy that property?

Perhaps, equivalent distances ? I'm just guessing.
 
  • #12
I don't know that. Sorry. Or, I would say: one is locally majorized by the other.
 
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  • #13
No problem,
it was not important. I was just curious. As I mentioned you already fully answered the main question.

Thanks!
 

1. What are quaternions and unit-spheres?

Quaternions are a type of mathematical object that extends the complex numbers and are used to represent rotations in three-dimensional space. Unit-spheres, also known as unit quaternions, are a set of quaternions with a magnitude of 1, which are used to represent rotations.

2. How are quaternions and unit-spheres different from other types of numbers and shapes?

Quaternions are different from other types of numbers because they have four components (a, b, c, d) instead of just two (a, b) like complex numbers. Unit-spheres are different from other shapes because they have a constant magnitude of 1, which makes them useful for representing rotations.

3. What are some applications of quaternions and unit-spheres?

Quaternions and unit-spheres have many applications in fields such as computer graphics, robotics, and physics. They are used to represent and manipulate 3D rotations and orientations, which are essential in these fields.

4. How do you perform operations on quaternions and unit-spheres?

To perform operations on quaternions and unit-spheres, you can use quaternion algebra, which is similar to complex number algebra. Addition, subtraction, multiplication, and division can all be performed on quaternions and unit-spheres.

5. Are there any disadvantages to using quaternions and unit-spheres?

One disadvantage of using quaternions and unit-spheres is that they can be more complex and difficult to understand compared to other types of numbers and shapes. They also require more storage space and may be less efficient to use in certain applications.

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