As you suspected, this is well known. At the moment, I can't get hold of references other than my own, but this has been known long before:

Salzmann, Betten, Grundhoefer, Haehl, Loewen, Stroppel: Compact Projective Planes, de Gruyter 1995. See in particular Chapter 1, where the relationship of quaternions and the orthogonal groups SO3 and SO4 is explained. 

Reality quaternion descriptions proven in studies of physical ( optical ) properties of the formation of the real and imaginary images of optical glass ball.

Gennady S. Melnikov Space-time fractal unity 

https://www.researchgate.net/publication/271486882_..______._Gennady_S._Melnikov_Space-time_fractal_unity

Оf the physical examination was the author  shows alarms equation geometric field of spatial frequency that is phase supplement symbol K. Maxwell's equations of electrodynamics

Book Г.С. Мельников Фрактальное единство пространства и времени. ...

In some sense this is the well known way to understand the quaternions like the complex numbers: The group SO_2 consists of real Matrices  (a, -b \\ b,a)  with  a^2+b^2 = 1 (*). If you omit (*), you receive C as a sub-algebra of M(2,R) (real 2x2 matrices). Likewise, the group SU_2 consists of complex matrices  (a, -b' \\ b, a')  where b' = \bar b (complex conjugate of b), with the condition |a|^2+|b|^2 = 1 (**). If you omit (**), you obtain  H  as a sub-algebra of  M(2,C). In your construction you use SO_3 in place of SU_2, but by means of the 2\pi-rotation (considered as different from id) you are working with its double cover which is SU_2.

Best regards

Jost

This is indeed well known, and in fact easy to see. For 

q = a + bI + cJ  + dK,  let \bar q = a -bI -cJ - dK. 

Then  |q|^2 = q\bar q = \bar q q = a^2 + b^2 + c^2  + d^2 is the standard Euclidean metric. Moreover \bar (ab) = \bar b \bar a.

Let E = im (H) = { q \in H  | \bar q = - q  i.e. q  = b I + cJ + d K} 

Hence  E is a 3 dimensional Euclidean space  and |x|^2 = -x^2 for x \in E. 

For all q \in H and all x \in E we have q x\bar q \in E because  

\bar (q x \bar q) = \bar\bar q \bar x \bar q = q (-x) \bar q = - (q x \bar q),

so  H acts on E, and it is easy to see that each quaternion acts as a linear transformation. Moreover if |q|^2 =  \bar q q=1, then 

|qx\bar q|^2 = - q x\bar q qx\bar q= -q x^2 \bar q = - x^2 q \bar q = -x^2 = |x|^2

so the Group of unit quaternions Sp(1) act as orthogonal transformations on E. In fact they act as rotations because the group of unit quaternions is a 3-sphere hence connected and so each induced orthogonal transformation has to have determinant 1.  

Dear Jost and Rogier,

             I mean a geometrical structure,

in euclidean three dimensional space,

isomorphic to usual numerical quaternions.

I have found it.

They are the usual rotations in the usual space,

with a given point (origin),

endowed with a given verse of rotation.

Dear Giancarlo,

the fact of the matter is that while your intuition is right on track, technically what you are describing does not quite work because you cannot quite keep track of the sense of the rotation once the rotation is over 2pi, especially if the rotations don't have an axis.

The scaling of the (non zero) quaternions is not really what causes the problems, although it is n fact slightly more natural (but not strictly necessary) to have the norm of the quaternion correspond to to scaling with the square in R^3. So lets concentrate on the unit quaternions and rotations.

What you really seem to want to do describe is to consider homotopy classes [Rt] of pathes with fixed endpoint starting in the identity In other words you have to keep track of a rotation of a frame and how you got there starting from a fixed frame, but if two such pathes to the same endpoint can be deformed into each other keeping the endpoint fixed, they have to be identified. This is a technical implementation of your idea of keeping track of the sense of orientation, and it has the great advantage that it is obvious how to compose them: [Rt][R't] = [Rt R't] , and likewise how to take the inverse. You may have seen this construction before in a course on topology because it is the standard construction of the universal covering. The universal covering of a (Lie) group is naturally a group and for SO(3) it is called Spin(3). As we have seen it comes with a group homomorphism Spin(3) --> SO(3) such that inverse image of the identity is discrete (in fact is a covering, the invers image of a small neighborhood is a copy of this neighborhood for each element of the fundamental group \pi_1 of the group. One can show that for SO(3) this means that Spin(3) --> SO(3) is a 2:1 covering)

Now unfortunately while this construction of the universal covering is very natural and universal it is also rather abstract. Fortunately the universal covering is universal in the sense that for any simply connected group S together with a covering homomorphism S --> SO(3), S is isomorphic to Spin(3) up to unique isomorphism. So what the unit quaternions Sp(1) do is provide you with such a map (by the construction I gave above) and that gives you an isomorphism of Spin(3) with Sp(1). This is because Sp(1) is obviously a 3-sphere and the 3-Sphere is simply connected.

Factoid1:

In fact this description is so convenient and explicit that the unit quaternions have become the standard representation of rotations in computer graphics especially video games. In computer games you have the very real need to find a path from the identity to a rotation so "the camera and your eye can follow ho you got there", and this works much better if the shortest path to a rotation/quaternion is only slightly different for slightly different rotations/quaternions.

Factoid2

The explicit identification Spin(3) = Sp(1) --> SO(3) is the easiest way to see that the covering is 2:1, and the fundamental group pi_1(SO(3)) = Z/2Z. It then gives the fundamental group pi_1(SO(n)) = Z/2Z for all n > 3 via standard theorems: the fibration SO(n) >--> SO(n+1) -->> S^n and the corresponding long exact sequence of groups

pi_2(S^n) --> pi_1(SO(n)) --> pi_1(SO(n+1)) --> pi_1(S^n)

Dear Rogier,

        thank you for your interesting presentation

of important well known facts about homotopy.

But I insist to say that the "numerical" structure

of unitary quaternions is also isomorphic

to the structure of rotations (of the three dimensional space)

endowed with a verse of rotation.

I agree with you that it is important to underly

that the constant unitary quaternions  1  and  -1

correspond to the identity rotaion and

to the rotαtion of  2π  with respect

to an arbitrary axis.

Best

Hi Giancarlo,

Maybe it would help to consider a formulation that explicitly relates quaternions to stretch rotations.  Any unit quaterion can be written as

q = cos(t/2) + sin(t/2)*(v_1 I + v_2 J + v_3 K),

where v=(v_1 I + v_2 J + v_3 K) is a unit-length vector, i.e., v_1^2+v_2^2+v_3^2=1.  It is well-known that for any vector x=(x_1 I + x_2 J + x_3 K), the quaternion product q*x*q' is also a vector (i.e., its scalar part is zero), as Rogier already showed.  Furthermore, q*x*q' is the simple rotation of x around axis v by angle t.  This is a classical result.  In kinematics, the representation of rotations as a unit-length 4-vector [e_0,e_1,e_2,e_3], with e_0=cos(t/2), e_i=sin(t/2)v_i, i=1,2,3, is known as Euler parameters of rotation.  Alternatively, for nonzero e_0, one can choose to parameterize rotation as [1,r_1,r_2,r_3]=[1,e_1/e_0,e_2/e_0,e_3/e_0], i.e., r_i=tan(t/2)v_i, which are known as Rodrigues parameters.  Clearly, the Rodrigues parameters have singularities at t=k*\pi, k any odd integer, which is why the unit quaternion representation is preferred and is ubiquitous in robotics and computer graphics.  I mention Rodrigues parameters only as a reference point for you to look up in the classical literature.  Euler and Rodrigues developed their treatments of rotation before Hamilton invented quaternions, but now the relationship between these is well known.

It is easy to see from the expression q*x*q' that -q and q give the same rotation, which they must since a rotation of 2\pi about any axis brings a body back to its initial orientation.  However, if we track Euler parameters by continuation while a body rotates about a fixed axis, v, it takes a rotation of t=4\pi to bring the parameters back to their initial values.  This is the double cover of the rotation group that Rogier already explained.

If we consider all quaternions, not just the unit length ones, q*x*q' becomes a stretch rotation with lengths multiplied by q*q', the squared length of q.  Alternatively, we can represent rotations as (q*x*q')/(q*q'), which can be a useful expression as long as the quaternion is not zero.

Hi Charles,

        thanks for your known and unknown (at least for me)

information you give to me.

But my problem is to ask you if the isomorphism

(not simply surjective map)

between unitary quaternions and

rotations of the ordinary space

endowed with a verse

is a well known fact 

@Giancarlo: I still don't understand what you mean by "verse". Just a sign does not work because SO(3) \times Z/2Z  is not a 3 sphere like the unit quaternions.

@Rogier: I simply mean, looking at a plane from one of the two semispaces,

clockwise or counterclockwise.

@Rainer Löwen:

Yiou say: "As you suspected, this is well known.

At the moment, I can't get hold of references".

Please.

Find a reference.

Just one reference.

Giancarlo, my problem is that your assertion is rather vague. When I said it's well-known, I obviously did not understand what exactly you mean, and frankly, I still don't. You need to be more explicit and formal. (E.g.: What do you mean by isomorphic --- the notion of isomorphism depends on which structure you are considering). Anyway, I don't think I have a reference for whatever your assertion is. Sorry.

Dear Rainer,

        believe me,

my assertion is completely clear and not vague.

Suppose we have a fixed plane

and we look at the plane from one of the two semispaces.

For every rotation of the plane

we may choose one of the two verses,

clockwise or counterclockwise,

In this way we have the concept

of oriented rotation

defined as a rotation endowed with a verse.

I say that the structure of unitary quaternions

is isomorphic to the structure of oriented rotations.

Thats all.

I think that your observations are well known to people familiar with quaternions, but I do not know a specific reference.  Perhaps this is just common folklore that many people know, but is not worth publishing in a separate paper.

Dear Ronald,

          if it is well known,

it should be written

in some book

or in some article,

I believe.

Perhaps you can write it down somewhere, but you run the risk of the referee saying that this result is too obvious to publish.  Good luck.

Giancarlo, you know you want to write it down! Do it for yourself first. And if the referee says like Ronald says, publish it for students, or in a teachers magazine or on the web!

Even if it is written somewhere, maybe you can make a better version !

Dear Christian and Ronald

Dear Giancarlo,

I made a little QuickTime movie from Geogebra 3D using a unit vector and varying each component of a quaternion to see associated rotation of the vector. Would be nicer with a plane curve like an ellipse (maybe later). Remains to proove preservation of group structures.

Theory taken from https://fr.wikipedia.org/wiki/Quaternions_et_rotation_dans_l'espace#Comparaisons_de_performances_avec_d.27autres_m.C3.A9thodes_de_rotation

Только

researchgate gennady s melnikov

http://ns1.npkgoi.ru/r_1251/investigations/fractal_opt/data6/data6.html

gmbh_lap_publition_978-3-659-59837

-1 (2).PDF

(PDF) Г.С. Мельников Фрактальное единство пространства и ...

https://www.researchgate.net/.../271486882_GS_Melnikov_Fraktalnoe_ edinstvo_prostranstva_i_vremeni_Gennady_S_Melnikov_Space-time_fract...

📷 Г.С. Мельников Фрактальное единство пространства и времени. Gennady S. Melnikov Space-time fractal unity. Book · October 2014 with 355 Reads.

Thanks Mr Melnikov. Will try to read with translator