On quaternionic product on some sets of hyperbolas

The aim of this paper is to introduce some products, induced by the quaternionic product, on the set of equilateral hyperbolas. The study is a mix of elements from Euclidean and projective geometry. Some properties of these products are highlighted and are connected with some special numbers as roots or powers of the unit. Then we extend these products in a natural manner to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas, using the algebra of octonions. Further, using an inversion, we extend these products to Bernoulli lemniscates and to Bernoulli q-lemniscates. Finally, using some isomorphisms, we extend these products to conics, and using an inversion, we extend these products to other curves. In order to highlight the importance of these studied products, the paper ends with some applications. 2010 Mathematics Subject Classification: 51N20, 51N14, 11R52, 11R06.


Introduction
Using the well-known product of quaternions we introduce some products on the set of equilateral hyperbolas considered in a projective way and we give some extensions of them. We define a first product, denoted ⊙ c ; then, as the c-square of the unit hyperbola H(1) : x 2 − y 2 − 1 = 0 is the degenerate hyperbola H(0) : x 2 − y 2 = 0, we introduce a second product, denoted ⊙ pc , to avoid the degeneration. A detailed study of both these products is the content of section 1. By looking to examples as well as to roots/powers of the unit 1 ∈ R we obtain some remarkable numbers, some of them algebraic but other of difficult nature.
Using an inversion, we extend the above products from the set of equilateral hyperbolas to sets of Bernoulli lemniscates and q-lemniscates, thus some interesting results are obtained.
Then, using a larger set of conics Q Γ 0 , we extend the above products and we obtain a field isomorphic to the field of complex numbers. For some particular values of the parameters, we obtain the product of equilateral hyperbolas firstly considered. Moreover, we extend the product on conics from Q Γ 0 to other curves.
Starting from the expression of an octonion as a pair of quaternions, we introduce a product of pairs of equilateral hyperbolas. While the products introduced on the set of equilateral hyperbolas are commutative, the considered product of pairs of equilateral hyperbolas is not.
Finally we propose three applications of the given products: the first two of them regard hyperbolic objects, namely the reduced equilateral hyperbola and hyperbolic matrices, concerning with multi-valued mapsand the third one returns to the Euclidean plane geometry and defines a chain of labels for a given polygon.
The present study is the hyperbolic counter-part of a similar work concerning circles in [5] while a more general Clifford product for EPH-cycles is introduced in [4] and it is a natural continuation of [5] due to the Lambert's and Riccati's analogies between the circle and the equilateral hyperbola, exposed in [2] and published as [3].

Quaternionic product of hyperbolas
Let us consider a given equilateral hyperbola H in the Euclidean plane with coordinates (x, y): We identificate H with a quaternion: The quaternion q(H) is pure imaginary if and only if the origin O(0, 0) belongs to H. Let us point out that the given hyperbola is expressed in a projective manner since the coefficient of the quadratic part is chosen as being 1. Hence the set of equilateral hyperbolas is a 3dimensional projective subspace of the 5-dimensional projective space of conics. Our study will be a mix of elements from Euclidean and projective geometry.
For H i , i = 1, 2 given by (a i , b i , c i ) we introduce a product of equilateral hyperbolas by: and we have: which gives a commutative expression only for the free term.
We restrict our study to equilateral hyperbolas H(r) already centered in O (in the chosen projecive setting); hence their set is a 1-dimensional projective subspace of the projective spaces considered above. For such a hyperbola we have: and hence the equation (4) yields: From the properties of quaternionic product we have that the above product can be also expressed in matrix product manner: Takeing into account the above considerations, we introduce the product law: Then we define on the set M = (0, +∞) a non-internal composition law : thus we have: Property 1.1 The product ⊙ c is commutative and associative but does not have a neutral element: In particular, the unit hyperbola is the square root of the degenerate hyperbola, Property 1.4 Concerning the squares we have: and the first relation (15) means that ⊙ c is a "shrinking" composition. The ⊙ c -square root of 1 is the number: while the ⊙ c -square root of c √ 1 is the number: Let us remark that c √ 1 is exactly the silver ratio Ψ := 1 + √ 2 and we point out that Ψ is a quadratic Pisot-Vijayaraghavan number considered as solution of: The conjugate of Ψ with respect to this algebraic equation is: Let us note that the radius involved in the well-known Hopf fibration as the Riemannian submersion S 3 (1) → S 2 ( 1 2 ) are involved in the expression of these two remarcable numbers, thus we intend to investigate this approach in a future study. Property 1.5 The given square (15) is: where the quaternion (2) has the Euclidean norm: We can avoid the degeneration (H(1)) 2 The product ⊙ pc is commutative with x ⊙ pc 1 = 1 and:

Quaternionic product of oriented hyperbolas
We extend the previous products from hyperbolas to oriented hyperbolas, that is pairs H := (H, ε) with ε ∈ {±1}. Property 2.1 Let (H i , ε i ), i = 1, 2 , be two oriented hyperbolas. Then: All the aspects and properties obtained above for hyperbolas remain also true in the case of oriented hyperbolas.

On quaternionic product on Bernoulli lemniscates and q-lemniscates
In [1] it is proved, using purely geometrical means, that the image of an equilateral hyperbola with foci F 1 and F 2 by an inversion I r with respect to the circle centered in O and with radius r = |OF 1 | = |OF 2 | is a Bernoulli lemniscate with the same foci F 1 and F 2 . We start this section with a complex approach in order to achieve easier the extension to the quaternionic approach.
Thus, we will prove the above assertion using complex numbers. Firstly, we associate to every point (x, y) in the Euclidean plane the complex number z = x+iy ∈ C. As z ′ = I r (z) = αz with α ∈ R * + and |I r (z)| · |z| = r 2 we have |αz| · |z| = r 2 , so α = r 2 |z| 2 = r 2 z · z and therefore the equation of the inversion I r is: The equation: of the equilateral hyperbola with foci ±a √ 2, 0 , taking into account that x 2 − y 2 = Re z 2 = z 2 + z 2 2 , can be written as: The image of the above equilateral hyperbola by the inversion I r has the equation As the equilateral hyperbola has the foci ±a √ 2, 0 while the Bernoulli lemniscate has the foci ± r 2 a √ 2 , 0 the foci are conserved by the inversion I r if and only if a More exactly, the image by the inversion I r of the equilateral hyperbola H r 2 2 that has the foci (±r, 0) is the Bernoulli lemniscate L(r) with the same foci, having the equation Let us note that for a Bernoulli lemniscate L(r) with parameter r > 0, the distance between the foci is 2r. We can introduce the products of Bernoulli lemniscates L(r 1 ) and L(r 2 ) in the same manner as the products of equilateral hyperbolas: All the properties proved for quaternionic products of equilateral hyperbolas are also true for quaternionic products of Bernoulli lemniscates.
A well-known property of an equilateral hyperbola given by equation (27) is expressed by: and, in the case of a Bernoulli lemniscate, given by the equation: is expressed by: These properties could be proved using pure geometric or analytic geometry means but we prove them using complex numbers.
Recall that the inversion I r with r = a √ 2 preserves the foci F the property ||MF 1 | − |MF 2 || = r √ 2 = const. and a current point M on its image by this inversion I r , the Bernoulli lemniscate L(r) given by the equation x 2 + y 2 2 = 2r 2 x 2 − y 2 , with the same foci, has the property |MF 1 | · |MF 2 | = |OF 1 | · |OF 2 | = r 2 = const.
We extend these constructions in a quaternionic setting. First we recall that M (x, y, z, w) ∈ R 4 has the quaternionic affix q = x + yi + z j + wk. We say that the hyperquadric in R 4 defined by the equation: H q (a 2 ) : is a q-equilateral hyperboloid. We can define its foci as the points ±a √ 2, 0, 0, 0 .
Also we say that: is the Bernoulli q-lemniscate with the points ± a √ 2 , 0, 0, 0 as foci.
In the following we will show that the names q-equilateral hyperboloid and Bernoulli q-lemniscate are fully justified because the main properties of equilateral hyperbola and Bernoulli lemniscate stated above in complex context are also preserved in quaternionic context.
Proposition 3.1 The relation (29), specific to a hyperbola, holds also true for a q-equilateral hyperboloid.
In a similar way with the quaternionic products on equilateral hyperbolas we can introduce the quaternionic products of q-equilateral hyperboloids H q (r 1 ) and H q (r 2 ): H q (r 1 ) ⊙ c H q (r 2 ) = H q (r 1 ⊙ c r 2 ), H q (r 1 ) ⊙ pc H q (r 2 ) = H q (r 1 ⊙ pc r 2 ).
Recall that the equilateral hyperbola (27) can be written in a complex form as (28) and, in an analogous way, the q-equilateral hyperboloid (32) can be written in a quaternionic form as: H q (a 2 ) : q 2 +q 2 = 2a 2 .
Analogously to the usual inversion (26) we can define a quaternionic inversion on R 4 \{0}: and it is easy to see that I 2 r := I r • I r = id thus, as in the case of the planar inversion, I r is an involution.

Proposition 3.2 The image of the q-equilateral hyperboloid (33) by the inversion I r is a Bernoulli q-lemniscate.
Since the q-equilateral hyperboloid has the foci ±a the inversion I r of the q-equilateral hyperboloid that has the foci (±r, 0, 0, 0), i.e. having the equation x 2 − y 2 − z 2 − w 2 = r 2 2 is the Bernoulli q-lemniscate with the same foci, having the equation x 2 + y 2 + z 2 + w 2 2 = 2r 2 x 2 − y 2 − z 2 − w 2 .

Proposition 3.3 The relation (31), specific to a Bernoulli lemniscate, holds also true for a Bernoulli q-lemniscate.
In a similar way with the quaternionic products of Bernoulli lemniscates we can introduce two quaternionic products of Bernoulli q-lemniscates: ∠ L q (r 1 ) ⊙ c L q (r 2 ) = L q (r 1 ⊙ c r 2 ), L q (r 1 ) ⊙ pc L q (r 2 ) = L q (r 1 ⊙ pc r 2 ).

On a quaternionic product of conics
We consider the set Q q 0 = {c + αq 0 ; c, α ∈ R}, where q 0 = ai + b j + dk is a pure imaginary quaternion, arbitrarily chosen but fixed, therefore a, b and d are fixed.
Since Q q 0 ⊂ Q is a vector subspace, generated by {1, q 0 }, we can consider also the additive group structure on Q q 0 . Moreover, the quaternionic product induces on Q * q 0 = Q q 0 \{0} a group structure isomorphic with the multiplicative group on C * . Therefore, using these two operations, Q q 0 is a field isomorphic with the field C. Thus we can identify Q q 0 with R 2 and even with C , using the isomorphism given by f : Now we consider a conic Γ in the Euclidean plane given by: Γ : x 2 + dy 2 + ax + by + c = 0 and we associate to Γ the quaternion q(Γ) = c + ai + b j + dk = (c, a, b, d) ∈ R 4 . Let Γ 1 and Γ 2 be two conics given by: We can associate a conic Γ 4 = Γ 1 ⊕Γ 2 corresponding to the sum and Γ 3 = Γ 1 ⊙ c Γ 2 corresponding to the product of the corresponding quaternions q 1 = q(Γ 1 ) and q 2 = q(Γ 2 ): where α 4 = α 1 + α 2 , c 4 = c 1 + c 2 and c 3 and α 3 are given by formulas (35). Thus, we can consider now the arbitrarily chosen but fixed conic Γ 0 : x 2 +dy 2 +ax+by = 0 and also the set of associated conics: ITM Web of Conferences 34, 03004 (2020) Third ICAMNM 2020 One can prove that ⊙ c is a commutative and associative law and has as neutral element the (imaginary) conic x 2 + 1 = 0 (corresponding to c = 1, α = 0). Let us note that for a conic Γ ∈ Q Γ 0 , the corresponding c and α are unique. Of course, a given conic can be seen as belonging to several families, but once the conical family is fixed, the corresponding c and α are unique; therefore the above operations on an arbitrary, but fixed family Q Γ 0 are well defined.
Property 4.1 The triple Q Γ 0 , ⊕, ⊙ c is a field isomorphic to the field of complex numbers.

Property 4.2
The corresponding group structure is isomorphic with the multiplicative circular group S 1 . Now let us consider α 1 c 2 + α 2 c 1 , α 1 , α 2 0. We obtain that the product of , therefore the product ⊙ c defined in the first section comes from the product ⊙ c,1 when ∆ 0 = 1 and it is restricted to the classes {[r, 1] ; r > 0}.
Thus if ∆ 0 = 1 then we can consider restrictions of the product ⊙ c,1 from P 1 = P 1 1 ∪ P 1 2 to P 1 1 or P 1 2 , where P 1 1 = {[r, 1] ; r ∈ R} and P 1 2 = {[1, r] ; r ∈ R}. We have to note that the products restricted to P 1 1 and P 1 2 are partial. Indeed, for example, if r ∈ R, then [r, 1] ⊙ c,1 [−r, 1] = −r 2 − 1, 0 = [1, 0] ∈ P 1 2 . One can explain now why ⊙ c does not have a neutral element when it is restricted to the classes {[r, 1] ; r > 0} or even to P 1 1 , since the neutral element [1, 0] ∈ P 1 2 does not belong to these sets. Notice also that the sum of parameters do not factorize to an additive law in the projective space P 1 .
More particulary, for α 1 = α 2 = 1, a = b = 0 and d = −1, we obtain the composition law associated to the family of equilateral hyperbolas approached in the first section.
Considering now the determinants: associated to a conic in a family Q Γ 0 , it is easy to see that the family does not always have only one type of conic. For example, in the case a = b = 0, d = −1, we have δ = α and ∆ = −c. If αc 0 then all the conics are non-degenerated with the center in origin; for α > 0 all the conics are hyperbolas; for α < 0 all the conics are ellipses; they are all real for c < 0 and all imaginary for c > 0.
ITM Web of Conferences 34, 03004 (2020) Third ICAMNM 2020 Thus the quaternionic product is a composition law that is internal on Q (a,b) and induces also an internal composition law on P (a,b) ( for every P x 1 , P x 2 ∈ P (a,b) we have P x 1 ⊙ c,∆ P x 2 = P x 3 ∈ P (a,b) , where ∆ = a 2 + b 2 ).

Property 4.3
The quaternionic product on Q (a,b) and the induced composition law on P (a,b) are commutative, associative and has as neutral element the quaternion, respectively the conic corresponding to (1, a, b, 0). 6 The quaternionic product on Q Γ 0 extended to other curves Let us consider a more general case, i.e. the following equation x 2 + dy 2 + c = 0, c 0. We have to analyze two different cases.
• If d < 0 then we have d = −µ 2 , so the equation x 2 − (µy) 2 + c = 0 is the equation of a hyperbola and can be written as (z + z) 2 + µ 2 (z − z) 2 + 4c = 0. Therefore, the image of this hyperbola by the inversion I r has the equation: For µ ±1 the equation (36) is x 2 + y 2 2 = r 4 −c x 2 − µ 2 y 2 which is (with the above discussion for c < 0, but also for c > 0) the equation of a generalized lemniscate.
Thus, if d < 0 for every type of above lemniscates L 1 , L 2 and L 3 , taking into account ⊙ c,∆ introduced in the previous section, we have L 1 ⊙ c,∆ L 2 = L 3 , where ∆ = µ 4 .
For α 1 = α 2 = 1 and µ = ±1 the above product has the same form as the product ⊙ c on the family of Bernoulli lemniscates, considered in a previous section.
• If d > 0 then we have d = µ 2 , so the equation is x 2 + (µy) 2 + c = 0, which is the equation of an ellipse when µ ±1 or of a circle when µ = ±1 and can be written as (z + z) 2 − µ 2 (z − z) 2 + 4c = 0. Therefore, the image of this curve (ellipse or circle) by the inversion I r has the equation: For µ = ±1 the equation (37) is x 2 + y 2 = r 4 −c , x 2 + y 2 0, which is the equation of a real circle (for c < 0 ) or an imaginary circle (for c > 0); for x 2 + y 2 = 0 the circle is degenerated in a point (the origin). Therefore, the image of the circle by the inversion I r is also a circle. Using ⊙ c,∆ introduced in a previous section, we have C 1 ⊙ c,∆ C 2 = C 3 , where ∆ = µ 4 and C 1 , C 2 and C 3 are circles.
For α 1 = α 2 = 1 and µ = ±1 we obtain the composition law associated to the above family of circles (see [5]). ITM Web of Conferences 34, 03004 (2020) Third ICAMNM 2020 For µ ±1 the equation (37) is x 2 + y 2 2 = r 4 −c x 2 + µ 2 y 2 , which is: -for c < 0, it is the equation of a Booth lemniscate (an oval of Booth with 0 as an isolated point, for µ 0, or a pair of externally tangent circles for µ = 0) or, -for c > 0, it is the equation of a curve degenerated in a double point.
Therefore for µ ±1, the image of the ellipse by the inversion I r is a Booth lemniscate or a curve degenerated in a point.
Let us note that if d = 0 then the equation x 2 + ax + by + c = 0 of parabolic type form can be written as: (z + z) 2 + 2a(z + z) − 2b(z − z)i + 4c = 0. Therefore, the image of this curve by the inversion I r has the equation r 4 x 2 + r 2 (ax + by) x 2 + y 2 + c x 2 + y 2 2 = 0.
Let us note that the product ⊙ c is commutative, but the quaternionic product and the octonionic product ⊙ o are non-commutative.
As in a previous section, we can introduce an octonionic product on pairs of oriented hyperbolas with (ε 1 ε 3 , ε 2 ε 4 ) on the second slot.
When the first pair or the second pair is (H(1), H(1)) or when the unit hyperbola H(1) is on the first or on the second position of the first pair or/and of the second pair of hyperbolas, interesting results are obtained.

Applications
Conversely, knowing the c-chain or the pc-chain of a polygon P we can deduce the length of some of its sides.
More precisely, if a polygon P has n = 2k + 1 sides and a vanishing c-chain then P has all its sides of length 1, and if a polygon P has n = 2k sides and a vanishing c-chain then P has the odd sides of a lenght l and the even sides of the lenght 1 l . If a polygon P has a constant pc-chain (1, ..., 1) and n = 2k + 1 sides then P has at least k + 1sides of length 1, and if P has n = 2k sides then P has at least k sides of length 1. Thus knowing the c-chain or the pc-chain of a polygon P we know a set of relations involving the length of its sides.