Mobility spaces and geodesics for the n-sphere
aa r X i v : . [ m a t h . G T ] F e b MOBILITY SPACES AND GEODESICS FOR THEN-SPHERE
J. P. FATELO AND N. MARTINS-FERREIRA
Abstract.
We introduce an algebraic system which can be usedas a model for spaces with geodesic paths between any two of theirpoints. This new algebraic structure is based on the notion ofmobility algebra which has recently been introduced as a modelfor the unit interval of real numbers. We show that there is astrong connection between modules over a ring and affine mobilityspaces over a mobility algebra. However, geodesics in general fail tobe affine thus giving rise to the new algebraic structure of mobilityspace. We show that the so called formula for spherical linearinterpolation, which gives geodesics on the n-sphere, is an exampleof a mobility space over the unit interval mobility algebra.
February 5, 2021; 01:51:471.
Introduction
The purpose of this work is to introduce an algebraic system whichcan be used to model spaces with geodesics. The main idea stemsfrom the interplay between algebra and geometry. In affine geometrythe notion of affine space is well suited for this purpose. Indeed, inan affine space we have scalar multiplication, addition and subtractionand so it is possible to parametrize, for any instant t ∈ [0 , x and y with the formula (1 − t ) x + ty . Such aline is clearly a geodesic path from x to y . In general terms, we mayuse an operation q = q ( x, t, y ) to indicate the position, at an instant t ,of a particle moving in a space X from a point x to a point y . Ifthe particle is moving along a geodesic path then this operation mustcertainly verify some conditions. The aim of this project is to presentan algebraic structure, ( X, q ), with axioms that are verified by anyoperation q representing a geodesic path in a space between any twoof its points. Mathematics Subject Classification.
Primary 08A99, 03G99; Secondary20N99, 08C15.
Key words and phrases.
Mobility algebra, mobi algebra, mobility space, affinespace, affine mobility space, unit interval, ternary operation, geodesic path,geodesics, sphere, n-sphere, Slerp, damped harmonic oscillator, projectiles.This work is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT)and Centro2020 through the Project references: UID/Multi/04044/2019; PAMI -ROTEIRO/0328/2013 (N º The first results concerning this investigation were presented in [6]and [7]. In [8], the first version of this text, mobility algebras (or mobialgebras) and mobility affine spaces are considered in more details.The remaining study of mobility spaces (mobi space for short) alreadypresented in [8] has been reformulated and further developed here.In the preprint [8], it is shown that every space with unique geodesicscan be given a mobi space structure. However, when geodesics are notunique (for instance when connecting antipodal points on the sphere)it is still possible to define a mobi space structure on that space. Thisis done by making appropriate choices and it is illustrated in the lastsection of this paper. The example of the sphere is considered withspherical linear interpolation (Slerp) whose formula gives rise to a mobispace structure. 2.
Mobi algebra
In this section we briefly recall the notion of mobi algebra, introducedin [7], and some of its basic properties. Several examples of differentnature were presented in [7] and [8]. As we will see in the next section(while introducing the notion of mobi space) a mobility algebra playsthe role of scalars (for a mobility space) in the same way as a ring (ora field) models the scalars for a module (or a vector space) over thebase ring (or field).In order to have an intuitive interpretation of its axioms we may con-sider a mobi algebra as a mobi space over itself and use the geometricintuition provided in section 3. Namely, that the operation p ( x, t, y )is the position of a particle moving from a point x to a point y at aninstant t while following a geodesic path. Definition 2.1. [7]
A mobi algebra is a system ( A, p, , ⁄ , , in which A is a set, p is a ternary operation and , ⁄ and are elements of A,that satisfies the following axioms: (A1) p (1 , ⁄ ,
0) = ⁄ (A2) p (0 , a,
1) = a (A3) p ( a, b, a ) = a (A4) p ( a, , b ) = a (A5) p ( a, , b ) = b (A6) p ( a, ⁄ , b ) = p ( a, ⁄ , b ) = ⇒ b = b (A7) p ( a, p ( c , c , c ) , b ) = p ( p ( a, c , b ) , c , p ( a, c , b )) (A8) p ( p ( a , c, b ) , ⁄ , p ( a , c, b )) = p ( p ( a , ⁄ , a ) , c, p ( b , ⁄ , b )) . Some properties of mobi algebras can be suitably expressed in termsof a unary operation ”()” and binary operations ” · ”, ” ◦ ” and ” ⊕ ”defined as follows (see [7] for more details). Definition 2.2.
Let ( A, p, , ⁄ , be a mobi algebra. We define: a = p (1 , a,
0) (1) a · b = p (0 , a, b ) (2) a ⊕ b = p ( a, ⁄ , b ) (3) a ◦ b = p ( a, b, . (4)Some properties of a mobi algebra follow. If ( A, p, , ⁄ ,
1) is a mobialgebra, then: ⁄ = ⁄ (5) a · ⁄ = ⁄ · a = 0 ⊕ a (6) ⁄ · a = ⁄ · a ′ ⇒ a = a ′ (7) p ( a, ⁄ , a ) = ⁄ (8) a = a ⇒ a = ⁄ (9) p ( a, b, c ) = p ( a, b, c ) (10) p ( c, b, a ) = p ( a, b, c ) (11) a ◦ b = b · a (12) ⁄ · p ( a, b, c ) = ( b · a ) ⊕ ( b · c ) . (13)We end this section recalling that if ( A, p, , ⁄ ,
1) is a mobi algebrathen ( A, ⊕ ), with ⊕ defined in (3), is a midpoint algebra. A midpointalgebra ([4], see also [9]) consists of a set and a binary operation ⊕ satisfying the following axioms:(idempotency) x ⊕ x = x (14)(commutativity) x ⊕ y = y ⊕ x (15)(cancellation) x ⊕ y = x ′ ⊕ y ⇒ x = x ′ (16)(mediality) ( x ⊕ y ) ⊕ ( z ⊕ w ) = ( x ⊕ z ) ⊕ ( y ⊕ w ) . (17)3. Mobi space
In this section we give the definition of a mobi space over a mobialgebra. Its main purpose is to serve as a model for spaces with ageodesic path connecting any two points. It is similar to a module overa ring in the sense that it has an associated mobi algebra which behavesas the set of scalars. In Section 5 we show that the particular case ofaffine mobi space is indeed the same as a module over a ring whenthe mobi algebra is a ring. In the last section, we present examplesof geodesics on the n -sphere and on an hyperbolic n -space as mobilityspaces over the unit interval. Definition 3.1.
Let ( A, p, , ⁄ , be a mobi algebra. An A -mobi space ( X, q ) , consists of a set X and a map q : X × A × X → X such that: (X1) q ( x, , y ) = x (X2) q ( x, , y ) = y J. P. FATELO AND N. MARTINS-FERREIRA (X3) q ( x, a, x ) = x (X4) q ( x, ⁄ , y ) = q ( x, ⁄ , y ) = ⇒ y = y (X5) q ( q ( x, a, y ) , b, q ( x, c, y )) = q ( x, p ( a, b, c ) , y )The axioms (X1) to (X5) are the natural generalizations of axioms (A3) to (A7) of a mobi algebra. The natural generalization of (A8) would be q ( q ( x , a, y ) , ⁄ , q ( x , a, y )) = q ( q ( x , ⁄ , x ) , a, q ( y , ⁄ , y )) . (18)This condition, however, is too restrictive and is not in general verifiedby geodesic paths. That is the reason why we do not include it. When(18) is satisfied for all x , x , y , y ∈ X and a ∈ A , then we say thatthe A -mobi space ( X, q ) is affine and speak of an A -mobi affine space(see [8] and Subsection 4.5).If we write x ⊕ y instead of q ( x, ⁄ , y ) and consider the special caseof equation (18) when a = ⁄ then we get the usual medial law (17).As an illustration of the fact that the medial law does not hold truein general for geodesic paths, let us consider the example of the unitsphere. The midpoint a ⊕ b of two points a and b on the equator isagain on the equator. Midpoint of North Pole n and any point c onequator is on the 45 th parallel. But the geodesic midpoint of two pointson the 45 th parallel does not live on the 45 th parallel, but somewhat tothe North of it; the 45 th parallel is not a geodesic. So ( a ⊕ b ) ⊕ ( n ⊕ n )is on the 45 th parallel, but ( a ⊕ n ) ⊕ ( b ⊕ n ) is not, but is north of it.This phenomenon is an aspect of the Gaussian curvature of the sphere.Every affine space ( X, +) over a commutative field of scalars can beconsidered as an affine mobi space ( X, q ) with q ( x, t, y ) = (1 − t ) x + ty .At the end of this paper we give the formula for geodesics on the n -sphere in terms of a mobi space structure.Here are some immediate consequences of the axioms for a mobispace. Proposition 3.1.
Let ( A, p, , ⁄ , be a mobi algebra and ( X, q ) anA-mobi space. It follows that: (Y1) q ( y, a, x ) = q ( x, a, y ) (Y2) q ( y, ⁄ , x ) = q ( x, ⁄ , y ) (Y3) q ( x, a, q ( x, b, y )) = q ( x, a · b, y ) (Y4) q ( q ( x, a, y ) , b, y ) = q ( x, a ◦ b, y ) (Y5) q ( q ( x, a, y ) , ⁄ , q ( x, b, y )) = q ( x, a ⊕ b, y ) (Y6) q ( x, ⁄ , q ( x, a, y )) = q ( x, a, q ( x, ⁄ , y )) (Y7) q ( q ( x, a, y ) , ⁄ , q ( y, a, x )) = q ( x, ⁄ , y ) (Y8) q ( q ( q ( x, a, y ) , b, x ) , ⁄ , q ( x, b, q ( x, c, y )))= q ( x, ⁄ , q ( x, p ( a, b, c ) , y )) (Y9) q ( x, a, y ) = q ( y, a, x ) ⇒ q ( x, a, y ) = q ( x, ⁄ , y ) (Y10) q ( x, a, y ) = q ( x, b, y ) ⇒ q ( x, p ( a, t, b ) , y ) = q ( x, a, y ) , for all t . Proof.
The following proof of (Y1) , bearing in mind (1), uses (X5) , (X2) and (X1) : q ( x, a, y ) = q ( x, p (1 , a, , y )= q ( q ( x, , y ) , a, q ( x, , y ))= q ( y, a, x ) . (Y2) follows directly from (A1) and (Y1) . Beginning with (2), (Y3) is a consequence of (X5) and (X1) : q ( x, a · b, y ) = q ( x, p (0 , a, b ) , y )= q ( q ( x, , y ) , a, q ( x, b, y ))= q ( x, a, q ( x, b, y )) . Considering (12), property (Y4) follows from (Y1) and (Y3) . q ( q ( x, a, y ) , b, y ) = q ( y, b, q ( y, a, x ))= q ( y, b · a, x )= q ( y, a ◦ b, x )= q ( x, a ◦ b, y ) . Considering (4), (Y5) is just a particular case of (X5) . To prove (Y6) ,we use (X5) , (6) and (X1) . q ( x, ⁄ , q ( x, a, y )) = q ( q ( x, , y ) , ⁄ , q ( x, a, y ))= q ( x, p (0 , ⁄ , a ) , y )= q ( x, p (0 , a, ⁄ ) , y )= q ( q ( x, , y ) , a, q ( x, ⁄ , y ))= q ( x, a, q ( x, ⁄ , y )) . The following proof of (Y7) is based on (Y1) , (X5) and (8); q ( q ( x, a, y ) , ⁄ , q ( y, a, x )) = q ( q ( x, a, y ) , ⁄ , q ( x, a, y ))= q ( x, p ( a, ⁄ , a ) , y )= q ( x, ⁄ , y ) . To prove (Y8) , we start with the important property (13) of the un-derlying mobi algebra and then get: q ( x, ⁄ · p ( a, b, c ) , y ) = q ( x, b · a ⊕ b · c, y ) ⇒ q ( x, p (0 , ⁄ , p ( a, b, c )) , y ) = q ( x, p ( b · a, ⁄ , b · c ) , y ) ⇒ q ( x, ⁄ , q ( x, p ( a, b, c ) , y )) = q ( q ( x, b · a, y ) , ⁄ , q ( x, b · c, y )) ⇒ q ( x, ⁄ , q ( x, p ( a, b, c ) , y ))= q ( q ( x, b, q ( x, a, y )) , ⁄ , q ( x, b, q ( x, c, y ))) . It is easy to see that ( Y9 ) is a direct consequence of ( Y7 ), while ( Y10 )is a consequence of (X3) and (X5) . (cid:3) J. P. FATELO AND N. MARTINS-FERREIRA
In order to have an intuition on the strength of the axioms for amobi-space, let us take a simple example with the variables x, y ∈ R and t ∈ [0 , q ( x, t, y ) = x cos ( t ) + y sin ( t )and observe that it satisfies q ( x, , y ) = x but not q ( x, , y ) = y . If weput q ( x, t, y ) = x cos (cid:16) t π (cid:17) + y sin (cid:16) t π (cid:17) then we have q ( x, , y ) = x and q ( x, , y ) = y but axiom (X3) , namely q ( x, t, x ) = x , fails for all values other than x = 0 or t ∈ { , } .If we change the map q to be q ( x, t, y ) = x cos (cid:16) t π (cid:17) + y sin (cid:16) t π (cid:17) (19)then we get axioms (X3) and (X4) but the axiom (X5) is not verified.Take for example, x = 1, y = 0, r = t = and s = 1 and observe that q ( x, r + t ( s − r ) , y ) = cos (cid:18) π (cid:19) while q ( q ( x, r, y ) , t, q ( x, s, y )) = cos (cid:16) π (cid:17) and they are not equal.One might expect that in order to fix this problem it would be suf-ficient to find a map θ such that q ( x, θ ( r + t ( s − r )) , y ) = q ( q ( x, θ ( r ) , y ) , θ ( t ) , q ( x, θ ( s ) , y )) . However, this is not so simple. Indeed, perhaps a first guess would beto consider the map θ ( t ) = π arcsin( t ). With this modification, (19)would become q ( x, θ ( t ) , y ) = x + ( y − x ) t . This new formula, however, still does not satisfy axiom (X5) .There is, nevertheless, a general procedure that leads to a mobi spaceout of the formula h ( x, t, y ) = x + ( y − x ) t , but it involves some extrawork. We have to introduce one extra dimension, while solving a certainsystem of equations (see Theorem 3.2 below).First we solve the system of two equations (cid:26) A + ( B − A ) r = xA + ( B − A ) s = y which has a unique solution for every x, y ∈ R , r, s ∈ R + and s = r ,namely (cid:18) AB (cid:19) = 1 s − r (cid:18) s − r − (1 − s ) 1 − r (cid:19) (cid:18) xy (cid:19) . (20) The mobi space on the set R × R + (over the unit interval) is thusgiven by the formula q (( x, r ) , t, ( y, s )) = ( h ( A, r + t ( s − r ) , B ) , r + t ( s − r ))with s = r and A , B obtained from equation (20). When s = r we put q (( x, r ) , t, ( y, s )) = ( x + t ( y − x ) , r ) . We end up with the operation q : ( R × R + ) × [0 , × ( R × R + ) → ( R × R + )defined by q (cid:0) ( x, r ) , t, ( y, s ) (cid:1) = (cid:16) x + ( y − x ) rt +( s − r ) t r + s , r + t ( s − r ) (cid:17) , if s = r (cid:0) x + t ( y − x ) , r (cid:1) , if s = r , (21)turning ( R × R + , q ) into a mobi space over the unit interval. Thisprocedure provides a way to construct examples of mobi spaces and itis detailed in the next theorem and generalized in Section 6. Furtherexamples are given in the next section. Theorem 3.2.
Consider two real valued functions f and g , of onevariable. Let I ⊆ R be an interval of the real numbers such that for any s, t ∈ I , with t = s , the following condition holds: f ( s ) g ( t ) = f ( t ) g ( s ) . (22) Let V be a real vector space and K : I → V be any function. Then, ( V × I, q ) is a mobi space over the mobi algebra ([0 , , p, , , , where q : ( V × I ) × [0 , × ( V × I ) → V × I is defined by the formula q (( x, s ) , a, ( y, t )) = ( f ( p ) A + g ( p ) B − K ( p ) , p ) , when t = s (23) with p ≡ p ( s, a, t ) = s + a ( t − s ) and A, B ∈ V the unique solutions ofthe system of equations (cid:26) f ( s ) A + g ( s ) B = x + K ( s ) f ( t ) A + g ( t ) B = y + K ( t ) , whereas q (( x, s ) , a, ( y, t )) = ( x + a ( y − x ) , s ) , when t = s. (24)The proof will be given in Proposition 6.2 from which Theorem 3.2 isjust a particular case.For the moment let us look at some examples and then compare thesingular case of affine mobi spaces with R-modules. J. P. FATELO AND N. MARTINS-FERREIRA Examples
In the following list of examples, unless otherwise stated, the under-lying mobi algebra structure (
A, p, , ⁄ ,
1) is the closed unit interval,i.e. A = [0 , , , p ( a, b, c ) = (1 − b ) a + bc, for all a, b, c ∈ A. We will refer to this algebra as the canonical mobi algebra . In eachcase, we present a set X and a ternary operation q ( x, a, y ) ∈ X , for all x, y ∈ X , and a ∈ A , verifying the axioms of Definition 3.1. We alsoexplain how it was obtained as an instance of some general construc-tion.4.1. The canonical mobi space. (1) Vector spaces provide examples. For instance: X = R n ( n ∈ N )and q ( x, a, y ) = (1 − a ) x + a y. (2) The well known technique of transporting the structure providesus with other ways of presenting the canonical structure. Forevery bijective map F : X → X , with X ⊆ R n , we get a mobispace ( X, q ) with q ( x, a, y ) = F − ((1 − a ) F ( x ) + a F ( y )) . For instance, in the case of dimension one:(a) If F ( x ) = log x and X = R + then we get the mobi space( X, q ), with q ( x, a, y ) = x − a y a . (b) If F ( x ) = x , then ( R + , q ) is a mobi space with q ( x, a, y ) = xyax + (1 − a ) y . Examples obtained directly from Theorem 3.2.
To apply Theorem 3.2, we need three functions f , g and K such that f ( s ) g ( t ) = f ( t ) g ( s ) for all s, t ∈ I, s = t . If g is a non-zero constantfunction, this condition just imply the injectivity of f . Let us beginwith K = 0, g = 1 and a function f injective in I .(1) With f : R + → R ; x x , we obtain X = R × R + with the formula q (( x, s ) , a, ( y, t )) = (cid:18) x + ( y − x ) 2 s a + ( t − s ) a t + s , s + a ( t − s ) (cid:19) . This is, in fact, the example displayed in equation (21). Notethat, since r, s ∈ R + , the second branch in (21) is not necessary.(2) With f : R + → R ; x x , we get the set X = R × R + with the formula q (( x, s ) , a, ( y, t )) = (cid:18) x + ( y − x ) a t (1 − a ) s + a t , s + a ( t − s ) (cid:19) . (3) With f : R → R ; x x , we can consider the set X = R and the formula q (( x, s ) , a, ( y, t )) = (cid:18) x + ( y − x ) 3 s a + 3 s ( t − s ) a + ( t − s ) a s + s t + t , s + a ( t − s ) (cid:19) , if ( s, t ) = (0 , q (( x, , a, ( y, x + a ( y − x ) , . (4) In general, applying Theorem 3.2 with g = 1, K = 0 and f aninjective real function of one variable, we get a mobi space inany set X ⊆ R for which the formula q (( x, s ) , a, ( y, t )) = (cid:18) x + ( y − x ) f ( s + ( t − s ) a ) − f ( s ) f ( t ) − f ( s ) , s + a ( t − s ) (cid:19) , (25)if s = t , and q (( x, s ) , a, ( y, s )) = ( x + a ( y − x ) , s ) , defines a map q : X × [0 , × X → X .At first glance, we could be skeptical that this operation q verify eventhe simplest properties of a mobi space, like (Y1) , for an arbitraryinjective function f , but it does. Indeed, for t = s , (25) imply q (( x, s ) , − a, ( y, t ))= (cid:18) x + ( y − x ) f ( t + a ( s − t )) − f ( s ) f ( t ) − f ( s ) , t + a ( s − t ) (cid:19) = (cid:18) x + ( y − x ) f ( t + a ( s − t )) − f ( t ) + f ( t ) − f ( s ) f ( t ) − f ( s ) , t + a ( s − t ) (cid:19) = (cid:18) y + ( x − y ) f ( t + a ( s − t )) − f ( t ) f ( s ) − f ( t ) , t + a ( s − t ) (cid:19) = q (( y, t ) , a, ( x, s )) . We will now see some examples obtained from physics.
Examples with physical interpretation.
The following examples, from classical mechanics, can be viewed asan application of Theorem 3.2 with specific expressions of f , g and K .(1) Consider a constant acceleration motion, with x ∈ R n , and thefollowing position equation x ( t ) = x + v t − k t . We can think, for instance, of a projectile motion in the plane R where k would be (0 , g ) with g being the gravitational accel-eration near the Earth’s surface. The constants x and v corre-spond to the usual initial conditions x (0) = x and x ′ (0) = v .Trying to construct a mobi space out of this context, withoutextra dimension, we could think of imposing boundary condi-tions like x (0) = x and x (1) = x , leading to: x ( t ) = x + ( x − x ) t + k t (1 − t ) . But then the operation q defined as q ( x , t, x ) = x ( t ) is not amobi operation: in particular, idempotency q ( x, t, x ) = x is notverified because a body could go up vertically and then downback to the same place; axiom (X5) is not verified either. Theway to go is to let the variable t flow freely in an extra dimensionwith boundary conditions like x ( t ) = x and x ( t ) = x . Theseconditions lead to: x ( t + a ( t − t )) = x + a ( x − x ) + k a (1 − a )( t − t ) . In the scope of Theorem 3.2, we could say that f ( t ) = t , g ( t ) = 1and K ( t ) = k t . For any k ∈ R n , we have then a mobi space( X, q ) over the canonical mobi algebra by taking the set X = R n +1 with the formula q (( x, s ) , a, ( y, t )) =( x + a ( y − x ) + k a (1 − a )( t − s ) , s + a ( t − s )) . (26)Remark: This example could be generalized to Special Rela-tivity [5]. However, the operation q is then a partial operationbecause, in Minkowski space-time, not every two points can bereached from one another if one point is not inside the light cone of the other.(2) The solutions for the one-dimension motion of the well-knowndamped harmonic oscillator are of the form Af ( t ) + Bg ( t ) − K ( t ) , where A and B are real parameters. If the oscillator is notdriven, K(t)=0. Depending on the circumstances, we can have:overdamping : f ( t ) = e α t , g ( t ) = e β t , α = β critical damping : f ( t ) = e α t , g ( t ) = t e α t underdamping : f ( t ) = e α t sin( β t ) , g ( t ) = e α t cos( β t ) , β = 0 , where α, β ∈ R depend on the oscillatory system. For the firsttwo cases, the condition (22) is verified for all s, t ∈ R , with t = s and therefore we can apply Theorem 3.2.(a) In the critical damping case and for any α ∈ R , we obtainthe following mobi space ( R , q ) over the canonical mobialgebra with the formula q (( x, s ) , a, ( y, t )) = (cid:0) (1 − a ) x e αa ( t − s ) + a y e α (1 − a )( s − t ) , s + a ( t − s ) (cid:1) . (b) In the overdamping case and for any α, β ∈ R , α = β ,we obtain the mobi space ( R , q ) over the canonical mobialgebra where q is defined, for t = s , by q (( x, s ) , a, ( y, t )) = (cid:16) e α (1 − a )( s − t ) − e β (1 − a )( s − t ) e αs + βt − e αt + βs e ( α + β ) t x + e βa ( t − s ) − e αa ( t − s ) e αs + βt − e αt + βs e ( α + β ) s y, s + a ( t − s ) (cid:17) , and, for t = s , by q (( x, s ) , a, ( y, s )) = ( x + a ( y − x ) , s ).(c) For the case of underdamping, we can still apply Theo-rem 3.2 if we restrict the possible values of s and t to, forinstance, I = [0 , π [. Here, we just mention that the casewhere f ( t ) = sin( βt ) and g ( t ) = cos( βt ) is analysed inSection 7.It is interesting to note that example (2a) can be obtained fromexample (2b) in the limit situation β → α .4.4. An example over a different mobi algebra.
So far we have considered examples of mobi spaces over theunit interval. Here is an example with a different mobi algebra.For the mobi algebra (
A, p, , ⁄ ,
1) let us use A = (cid:8) ( t , t ) ∈ R : | t | ≤ t ≤ − | t | (cid:9) ⁄ = (cid:18) , (cid:19) ; 1 = (1 ,
0) ; 0 = (0 , p ( a, b, c ) = ( a − b a − b a + b c + b c ,a − b a − b a + b c + b c ) . And for the mobi space (
X, q ): X = [0 ,
1] and, with h = ± q ( x, ( t, s ) , y ) = (1 − t − h s ) x + ( t + h s ) y. Affineness of the examples.
We end this section with some comments on whether the examplespresented verify the affine condition (18) or not. Examples like thosecorresponding to example 4.2(4) are, in general, not affine in the sensethat they don’t verify q (cid:18) q [( x , s ) , a, ( y , t )] , , q [( x , s ) , a, ( y , t )] (cid:19) = q (cid:18) q [( x , s ) , , ( x , s )] , a, q [( y , t ) , , ( y , t )] (cid:19) . Indeed, in example 4.2(1) for instance, we have that q (cid:18) q [(0 , , , (0 , , , q [(1 , , , (0 , (cid:19) = (cid:18) , (cid:19) but q (cid:18) q [(0 , , , (1 , , , q [(0 , , , (0 , (cid:19) = (cid:18) , (cid:19) . Similarly, in example 4.2(3), we have for instance: q (cid:18) q [(0 , , , (0 , , , q [(1 , , , (0 , (cid:19) = (cid:18) , (cid:19) while q (cid:18) q [(0 , , , (1 , , , q [(0 , , , (0 , (cid:19) = (cid:18) , (cid:19) . O course, the canonical mobi spaces 4.1 are affine. Examples 4.2(2),4.3(1) and 4.3(2a) correspond also to affine mobi spaces while 4.3(2b)does not. Indeed, for 4.3(2b), we have for instance that q (cid:0) q [(0 , , , (0 , , , q [(1 , , , (0 , (cid:1) = (cid:18) ( e α/ − e β/ )( e α/ + e β/ ) e α − e β , (cid:19) but q (cid:0) q [(0 , , , (1 , , , q [(0 , , , (0 , (cid:1) = (cid:18) e α + β ) / ( e − α/ − e − β/ )( e β/ − e α/ )( e α/ − e β/ )( e α − e β ) , (cid:19) . The two results are different if α = β . However, in the limit situationwhen β → α , the critical case is recovered and the two results arenaturally equal. The example 4.3(2c) is not affine either.In the next section, we compare affine mobi spaces [8] with modulesover a ring with one-half. Comparison with R-modules
Consider a unitary ring ( R, + , · , , R contains the inverse of 1 + 1, then it is a mobi algebra and if amobi algebra ( A, p, , ⁄ ,
1) contains the inverse of ⁄ , in the sense of(2), then it is a unitary ring. In this section, we will compare a moduleover a ring R with a mobi space over a mobi algebra A. First, let usjust recall that a module over a ring R is a system ( M, + , e, ϕ ), where ϕ : R → End( M ) is a map from R to the usual ring of endomorphisms,such that ( M, + , e ) is an abelian group and ϕ is a ring homomorphism.The following theorem shows how to construct a mobi space from amodule over a ring containing the inverse of 2. Theorem 5.1.
Consider a module ( X, + , e, ϕ ) over a unitary ring ( A, + , · , , . If A contains (1 + 1) − = ⁄ then ( X, q ) is an affinemobi space over the mobi algebra ( A, p, , ⁄ , , with p ( a, b, c ) = a + bc − ba (27) q ( x, a, y ) = ϕ − a ( x ) + ϕ a ( y ) . (28) Proof. ( A, p, , ⁄ ,
1) is a mobi algebra by Theorem 7.2 of [7]. We showhere that the axioms of Definition 3.1, as well as (18), are verified. Thefirst three axioms are easily proved: q ( x, , y ) = ϕ ( x ) + ϕ ( y ) = x + e = xq ( x, , y ) = ϕ ( x ) + ϕ ( y ) = e + y = yq ( x, a, x ) = ϕ − a ( x ) + ϕ a ( x ) = ϕ − a + a ( x ) = ϕ ( x ) = x. Axiom (X4) is due to the fact that ⁄ + ⁄ = 1 and consequently ϕ ⁄ ( y ) = ϕ ⁄ ( y ) ⇒ ϕ ⁄ ( y ) + ϕ ⁄ ( y ) = ϕ ⁄ ( y ) + ϕ ⁄ ( y ) ⇒ ϕ ( y ) = ϕ ( y ) ⇒ y = y . Next, we give a proof of Axiom (X5) . It is relevant to notice that,besides other evident properties of the module X , the associativityof + plays an important part in the proof: q ( q ( x, a, y ) , b, q ( x, c, y ))= ϕ − b ( ϕ − a ( x ) + ϕ a ( y )) + ϕ b ( ϕ − c ( x ) + ϕ c ( y ))= ϕ − b ( ϕ − a ( x )) + ϕ − b ( ϕ a ( y )) + ϕ b ( ϕ − c ( x )) + ϕ b ( ϕ c ( y ))= ϕ (1 − b )(1 − a ) ( x ) + ϕ b (1 − c ) ( x ) + ϕ (1 − b ) a ( y ) + ϕ bc ( y )= ϕ − a + ba − bc ( x ) + ϕ a − ba + bc ( y )= ϕ − p ( a,b,c ) ( x ) + ϕ p ( a,b,c ) ( y )= q ( x, p ( a, b, c ) , y ) . It remains to prove (18): q ( q ( x , a, y ) , ⁄ , q ( x , a, y ))= ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y )) + ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y ))= ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y ) + ϕ (1 − a ) ( x ) + ϕ a ( y ))= ϕ ⁄ ( ϕ − a ( x + x ) + ϕ a ( y + y ))= ϕ (1 − a ) ⁄ ( x + x ) + ϕ a ⁄ ( y + y )= ϕ (1 − a ) ( ϕ ⁄ ( x ) + ϕ ⁄ ( x )) + ϕ a ( ϕ ⁄ ( y ) + ϕ ⁄ ( y ))= ϕ (1 − a ) ( q ( x , ⁄ , x )) + ϕ a ( q ( y , ⁄ , y ))= q ( q ( x , ⁄ , x )) , a, q ( y , ⁄ , y )) . (cid:3) Theorem 5.2.
Consider an affine mobi space ( X, q ) , with a fixed cho-sen element e ∈ X , over a mobi algebra ( A, p, , ⁄ , . If A contains such that p (0 , ⁄ ,
2) = 1 then ( X, + , e, ϕ ) is a module over the unitaryring ( A, + , · , , , with a + b = p (0 , , p ( a, ⁄ , b )) (29) a · b = p (0 , a, b ) (30) ϕ a ( x ) = q ( e, a, x ) (31) x + y = q ( e, , q ( x, ⁄ , y )) = ϕ ( q ( x, ⁄ , y )) . (32) Proof. ( A, + , · , ,
1) is a unitary ring by Theorem 7.1 of [7]. We provehere that ( X, + , e, ϕ ) is a module over A . First, we observe that, usingin particular (Y3) of Proposition 3.1, we have: q ( e, ⁄ , x + y ) = q ( e, ⁄ , q ( e, , q ( x, ⁄ , y )))= q ( e, ⁄ · , q ( x, ⁄ , y ))= q ( e, , q ( x, ⁄ , y ))= q ( x, ⁄ , y ) . Then, the property (18) of an affine mobi space is essential to provethe associativity of the operation + of the module: q ( e, ⁄ , q ( e, ⁄ , ( x + y ) + z )) = q ( q ( e, ⁄ , e ) , ⁄ , q ( x + y, ⁄ , z ))= q ( q ( e, ⁄ , x + y ) , ⁄ , q ( e, ⁄ , z ))= q ( q ( x, ⁄ , y ) , ⁄ , q ( e, ⁄ , z ))= q ( q ( x, ⁄ , e ) , ⁄ , q ( y, ⁄ , z ))= q ( q ( e, ⁄ , x ) , ⁄ , q ( e, ⁄ , y + z ))= q ( q ( e, ⁄ , e ) , ⁄ , q ( x, ⁄ , y + z ))= q ( e, ⁄ , q ( e, ⁄ , x + ( y + z ))) Which, by (X4) , implies that ( x + y ) + z = x + ( y + z ). Commutativityof + and the identity nature of e are easily proved: q ( e, ⁄ , e + x ) = q ( e, ⁄ , x ) ⇒ e + x = xq ( e, ⁄ , x + y ) = q ( x, ⁄ , y ) = q ( y, ⁄ , x )= q ( e, ⁄ , y + x ) ⇒ x + y = y + x. Cancellation is achieved with − x = q ( e, p (1 , , , x ). Indeed: q ( e, ⁄ , q ( e, p (1 , , , x ) + x ) = q ( q ( e, p (1 , , , x ) , ⁄ , x )= q ( q ( e, p (1 , , , x ) , ⁄ , q ( e, , x ))= q ( e, p ( p (1 , , , ⁄ , , x )= q ( e, p (1 , p (2 , ⁄ , , , x )= q ( e, p (1 , , , x )= q ( e, , x ) = e = q ( e, ⁄ , e )To prove that ϕ a ( x + y ) = ϕ a ( x ) + ϕ a ( y ), we will again need (18): q ( e, ⁄ , ϕ a ( x + y )) = q ( e, ⁄ , q ( e, a, x + y ))= q ( q ( e, a, e ) , ⁄ , q ( e, a, x + y ))= q ( q ( e, ⁄ , e ) , a, q ( e, ⁄ , x + y ))= q ( e, a, q ( x, ⁄ , y ))= q ( q ( e, a, x ) , ⁄ , q ( e, a, y ))= q ( ϕ a ( x ) , ⁄ , ϕ a ( y ))= q ( e, ⁄ , ϕ a ( x ) + ϕ a ( y )) . To prove that ϕ a + b ( x ) = ϕ a ( x )+ ϕ b ( x ), let us first recall that, in a mobialgebra with 2 and a + b = p (0 , , p ( a, ⁄ , b )), we have the followingproperty: p (0 , ⁄ , a + b ) = p ( a, ⁄ , b ) . We then have q ( e, ⁄ , ϕ a + b ( x )) = q ( e, ⁄ , q ( e, a + b, x ))= q ( q ( e, , x ) , ⁄ , q ( e, a + b, x ))= q ( e, p (0 , ⁄ , a + b ) , x )= q ( e, p ( a, ⁄ , b ) , x )= q ( q ( e, a, x ) , ⁄ , q ( e, b, x ))= q ( ϕ a ( x ) , ⁄ , ϕ b ( x ))= q ( e, ⁄ , ϕ a ( x ) + ϕ b ( x )) . The last two properties are easily proved: ϕ a · b ( x ) = q ( e, a · b, x ) = q ( e, a, q ( e, b, x )) = ϕ a ( ϕ b ( x )) ϕ ( x ) = q ( e, , x ) = x. (cid:3) Proposition 5.3.
Consider a R-module ( X, + , e, ϕ ) within the condi-tions of Theorem 5.1 and the corresponding mobi space ( X, q ) . Thenthe R-module obtained from ( X, q ) by Theorem 5.2 is the same as ( X, + , e, ϕ ) .Proof. From (
X, q ), we define x + ′ y = q ( e, , q ( x, ⁄ , y )) and ϕ ′ a ( x ) = q ( e, a, x )and obtain the following equalities: x + ′ y = e + ϕ ( q ( x, ⁄ , y ))= ϕ ( ϕ ⁄ ( x ) + ϕ ⁄ ( y ))= ϕ · ⁄ ( x ) + ϕ · ⁄ ( y )= x + yϕ ′ a ( x ) = ϕ − a ( e ) + ϕ a ( x ) = e + ϕ a ( x ) = ϕ a ( x ) . (cid:3) Proposition 5.4.
Consider an affine mobi space ( X, q ) within theconditions of Theorem 5.2 and the corresponding module ( X, + , e, ϕ ) .Then the affine mobi space obtained from ( X, + , e, ϕ ) by Theorem 5.1is the same as ( X, q ) .Proof. From ( X, + , e, ϕ ), we define q ′ ( x, a, y ) = ϕ − a ( x ) + ϕ a ( y )and obtain the following equalities: q ′ ( x, a, y ) = q ( e, a, x ) + q ( e, a, y )= q ( x, a, e ) + q ( e, a, y )= q ( e, , q ( q ( x, a, e ) , ⁄ , q ( e, a, y ))) . Now, because we are considering that (
X, q ) is affine, we get: q ′ ( x, a, y ) = q ( q ( e, a, e ) , , q ( q ( x, ⁄ , e ) , a, q ( e, ⁄ , y )))= q ( q ( e, , q ( e, ⁄ , x )) , a, q ( e, , q ( e, ⁄ , y )))= q ( q ( e, · ⁄ , x ) , a, q ( e, · ⁄ , y ))= q ( x, a, y ) . (cid:3) We have completely characterized affine mobi spaces in terms ofmodules over a unitary ring in which 2 is invertible. In a sequel to thispaper we will investigate how to characterize mobi spaces in terms ofhomomorphisms between mobi algebras.We finish this section by taking a closer look to Example 4.3(1)and a related module. This is an example of an affine mobi spaceand can be extended to the case where the underlying mobi algebra is ( R , p, , , R , + , · , , R n +1 , + , , ϕ ) with x, y, k ∈ R n , s, t ∈ R and( x, s ) + ( y, t ) = ( x + y − kst, s + t ) ϕ a ( x, s ) = ( ax + ka (1 − a ) s , as ) . By Theorem 5.1, we can construct a mobi space from this module andverify that it is the same as (26). Of course, in this example, the moduleis a vector field and there is a homomorphism, namely f ( x, s ) = ( x + k ( s − s ) , s )from ( R n +1 , + , , ϕ ) to the usual vector field in R n +1 .In the following section we will thoroughly analyse a procedure toconstruct examples of mobi spaces which in general are not affine mobispaces. In a sequel to this work we will investigate the case of spaceswith geodesics and how to construct mobi spaces out of them.6. Non-affine mobi spaces
We present here a general result from which Theorem 3.2 can bededuced. In general, the examples that are obtained in this way arenot affine.
Proposition 6.1.
Let ( X, q X ) and ( Y, q Y ) be two mobi spaces over amobi algebra ( A, p ) . Suppose we have two sets U , V and a function h : U × Y × V → X such that the system (cid:26) h ( α, y , β ) = x h ( α, y , β ) = x (33) has a unique solution for every x , x ∈ X and any y , y ∈ Y with y = y , namely (cid:26) α = α ( x , y , x , y ) β = β ( x , y , x , y ) . (34) Then, ( X × Y, q ) is a mobi space over the mobi algebra ( A, p ) where q : ( X × Y ) × A × ( X × Y ) → ( X × Y ) is defined using the map χ , via (34), χ ( x , y , a, x , y ) = (cid:26) h [ α, q Y ( y , a, y ) , β ] if y = y q X ( x , a, x ) if y = y . (35) as q (( x , y ) , a, ( x , y )) = ( χ ( x , y , a, x , y ) , q Y ( y , a, y )) . Proof.
The axioms (X1) , (X2) and (X3) are direct consequences of(33) and the fact that q Y and q X are operations of mobi spaces. Toprove (X4) , we first observe that q [( x , y ) , ⁄ , ( x , y )] = q [( x , y ) , ⁄ , ( x ′ , y ′ )] (36) implies q Y ( y , ⁄ , y ) = q Y ( y , ⁄ , y ′ ) and hence y ′ = y . If y = y , wealso get q X ( x , ⁄ , x ) = q X ( x , ⁄ , x ′ ) and consequently x ′ = x . When y = y , y ′ = y and (36) imply h [ α, q Y ( y , ⁄ , y ) , β ] = h [ α ′ , q Y ( y , ⁄ , y ) , β ′ ] ≡ x , where α ′ = α ( x , y , x ′ , y ) and β ′ = β ( x , y , x ′ , y ). Now, because y = y ⇒ q Y ( y , ⁄ , y ) = y , the system (cid:26) h ( α, y , β ) = x h ( α, q Y ( y , ⁄ , y ) , β ) = x has a unique solution, we then conclude that α = α ′ and β = β ′ andconsequently that x ′ = h ( α ′ , y , β ′ ) = h ( α, y , β ) = x . Let us now prove (X5) . We have to prove that Q = Q where: Q ≡ q [( x , y ) , p ( a, b, c ) , ( x , y )] Q ≡ q (cid:16) q [( x , y ) , a, ( x , y )] , b, q [( x , y ) , c, ( x , y )] (cid:17) . To simplify the presentation of the proof, the following notations areused: y a = q Y ( y , a, y ) , y c = q Y ( y , c, y ) ,χ a = h ( α, y a , β ) , χ c = h ( α, y c , β ) . • Considering y = y and y a = y c , we have Q = ( h [ α, q Y ( y , p ( a, b, c ) , y ) , β ] , q Y [ y , p ( a, b, c ) , y ])= ( h [ α, q Y ( y a , b, y c ) , β ] , q Y [ y a , b, y c ])and Q = q h ( χ a , y a ) , b, ( χ c , y c ) i = ( h [ ˜ α, q Y ( y a , b, y c ) , ˜ β ] , q Y [ y a , b, y c ]) , where ˜ α and ˜ β are the unique solutions of the system (cid:26) h ( ˜ α, y a , ˜ β ) = χ a h ( ˜ α, y c , ˜ β ) = χ c which imply that ˜ α = α and ˜ β = β , by definition of χ a and χ c and because y a = y c , therefore Q = Q . • Considering y = y , and hence y a = y c = y , we have Q = (cid:16) q X [ x , p ( a, b, c ) , x ] , q Y [ y , p ( a, b, c ) , y ] (cid:17) = (cid:16) q X [ q X ( x , a, x ) , b, q X ( x , c, x )] , y (cid:17) and Q = q (cid:16) ( q X [ x , a, x ] , y a ) , b, ( q X [ x , c, x ] , y c ) (cid:17) = (cid:16) q X [ q X ( x , a, x ) , b, q X ( x , c, x )] , q Y ( y a , b, y c ) (cid:17) = (cid:16) q X [ q X ( x , a, x ) , b, q X ( x , c, x )] , y (cid:17) implying that Q = Q . • Considering y = y and y a = y c , hence χ a = χ c , we have Q = ( h [ α, q Y ( y a , b, y c ) , β ] , q Y [ y a , b, y c ])= ( χ a , y a )and Q = q h ( χ a , y a ) , b, ( χ c , y c ) i = (cid:16) q X [ χ a , b, χ c ] , q Y [ y a , b, y c ]) (cid:17) = ( χ a , y a )= Q . (cid:3) As an example, consider U = X = R , V = R + , Y = R +0 , ( A, p ) thecanonical mobi algebra and h ( α, y, β ) = α β y . Then, for y = y , h [ α ( x , y , x , y ) , t, β ( x , y , x , y )] = x t − y y − y x y − ty − y , and if t is q Y ( y , a, y ) = y + a ( y − y ), we get: q [( x , y ) , a, ( x , y )] = ( x − a x a , y + a ( y − y )) . This expression is well-defined even for y = y . This leaves no optionfor q X if we want a continuous operation, as the only possibility is q X ( x , a, x ) = x − a x a . But any other mobi operation is allowed when y = y and we can write: q [( x , y ) , a, ( x , y )] = ( q X ( x , a, x ) , y ) . This example compares with Example 4.1(2a). Note that in Example4.2(3), the branch corresponding to ( s, t ) = (0 ,
0) cannot be obtainedby continuity due to the fact that the limit ( s, t ) → (0 ,
0) does notexist. However, the canonical expression at (0 ,
0) is the choice whichcorresponds to approaching the origin through the path t = s .A useful particular case is when X is a vector space and h ( α, t, β ) = αf ( t ) + βg ( t ) for some scalar maps f and g . Note that Theorem 3.2 isa reformulation of the following proposition. Proposition 6.2.
Let ( X, q X ) and ( Y, q Y ) be two mobi spaces over amobi algebra ( A, p ) . Suppose moreover that X is a vector space over a scalar field F and let f : Y → F and g : Y → F be two functions suchthat, for any y , y ∈ Y with y = y , the following inequality holds f ( y ) g ( y ) = g ( y ) f ( y ) . (37) Furthermore, we consider a function K : Y → X . Then ( X × Y, q ) isa mobi space over ( A, p ) considering that q : ( X × Y ) × A × ( X × Y ) → ( X × Y ) is defined as q [( x , y ) , a, ( x , y )] = (cid:16) χ a ( x , y , x , y ) , q Y ( y , a, y ) (cid:17) with χ a ( x , y , x , y )= g ( y ) ( x + K ( y )) − g ( y ) ( x + K ( y )) f ( y ) g ( y ) − f ( y ) g ( y ) f [ q Y ( y , a, y )] − f ( y ) ( x + K ( y )) − f ( y ) ( x + K ( y )) f ( y ) g ( y ) − f ( y ) g ( y ) g [ q Y ( y , a, y )] − K [ q Y ( y , a, y )] , when y = y and χ a ( x , y, x , y ) = q X ( x , a, x ) otherwise.Proof. This is just Proposition 6.1 for the case h ( α, t, β ) = α f ( t ) + β g ( t ) − K ( t ) . With U = V = X , the system (33) simply reads f ( y ) g ( y ) f ( y ) g ( y ) ! αβ ! = x + K ( y ) x + K ( y ) ! . (cid:3) To illustrate this Proposition, Example 4.2(4) can be generalizedusing on Y (generalizing the set I ) an arbitrary mobi space. Consider h ( α, t, β ) = α f ( t ) + β , in any set X for which the next formula is welldefined. Then, when y = y , q [( x , y ) , a, ( x , y )] =( x + ( x − x ) f ( q Y ( y , a, y )) − f ( y ) f ( y ) − f ( y ) , q Y ( y , a, y )) . When y = y , q [( x , y ) , t, ( x , y )] = ( q X ( x , a, x ) , y ) for any mobioperation q X .Even when the system of equations (33) does not have a unique so-lution, then, in some cases, it is still possible to define a mobi-space.This will be illustrated with the formula for spherical linear interpola-tion giving geodesics on the n -sphere. Geodesics on the n-sphere
The purpose of this section is to show that a mobi space can beobtained using the geodesic curves on the n -sphere S n = { x ∈ R n +1 | h x, x i E = 1 } and on one sheet of the two-sheeted hyperbolic n -space [15], as forinstance H n = { x ∈ R n +1 | h x, x i L = − , x > } . The notations h , i E and h , i L are used for the usual Euclidean andLorentzian inner products, respectively. For the construction of themobi operation for both cases at once, it is convenient to consider thefamily of functions f ( a ) = e αa − e − αa α and g ( a ) = e αa + e − αa , (38)where a ∈ R and the parameter α is a non-zero complex number. Thesefunctions are real functions if and only if α is a real number or a pureimaginary number. In particular, we have that: • α = 1 ⇒ f ( a ) = sinh a and g ( a ) = cosh a ; • α = i ⇒ f ( a ) = sin a and g ( a ) = cos a ; • α → ⇒ f ( a ) = a and g ( a ) = 1.For our purpose, we want to consider only real functions and therefore,for the rest of this section, it is understood that α is such that f and g are real. In any case, the functions (38) verify the following properties: − α f ( a ) + g ( a ) = 1 (39) f ( a ) g ( b ) + f ( b ) g ( a ) = f ( a + b ) (40) α f ( a ) f ( b ) + g ( a ) g ( b ) = g ( a + b ) (41) f ( − a ) = − f ( a ) , g ( − a ) = g ( a ) (42) f (0) = 0 , g (0) = 1 . (43)In general terms, let us consider an interval I ∈ R containing 0 whereg is injective. Let V be an inner product space, with the inner productdenoted by h , i , and a subspace X ⊆ { x ∈ V | h x, x i = − α } . Innerproduct here means a nondegenerate symmetric bilinear form. We aregoing to show that when there exists a unique function θ : X × X → I such that − α g [ θ ( x, y )] = h x, y i , X can be given the structure of amobi space. For instance, if X is S n , θ may be defined as θ ( x, y ) = arccos( h x, y i E ) with I = [0 , π ]and if X = H n , θ may be defined as θ ( x, y ) = arccosh( −h x, y i L ) with I = [0 , + ∞ [ . The expressions are similar for any pure imaginary or non-zero realnumber α . The next two Propositions show explicitly how to constructa mobi operation on X using the functions f , g and θ . This con-struction is based on the spherical linear interpolation (Slerp) used incomputer graphics [14]. The first proposition is for the cases where thegeodesic between two points is unique which occur when the only zeroof f , in I , is zero. This is what happens for H n but not for S n becausesin( π ) = 0. Nevertheless, the Proposition 7.1 could still be applied to aportion of the n-sphere which does not contain antipodal points, suchas for example { x ∈ R n +1 | h x, x i E = 1 , x > } . Proposition 7.1.
Consider the real functions f and g , of one realvariable, verifying the properties (39) to (43) for some number α . Sup-pose that g is injective in an interval I containing and that, for a ∈ I ,we have f ( a ) = 0 ⇐⇒ a = 0 . Let ( V, h , i ) be a real inner vector space and consider a subspace X ⊆ { x ∈ V | h x, x i = − α } . If there exists a unique function θ : X × X → I such that h x, y i = − α g [ θ ( x, y )] (44) and θ ( x, y ) = 0 ⇐⇒ y = x, then ( X, q ) is a mobi space over thecanonical mobi algebra where the ternary operation q : X × [0 , × X → X is defined, for x = y , by q ( x, t, y ) = f [ θ ( x, y ) (1 − t )] f [ θ ( x, y )] x + f [ θ ( x, y ) t ] f [ θ ( x, y )] y, (45) and, otherwise, by q ( x, t, x ) = x .Proof. To simplify the presentation, we use the notation Ω ≡ θ ( x, y ).First, we have to prove that, when x, y ∈ X , q ( x, t, y ) is still in X . Thecase y = x is obvious. For y = x : h q ( x, t, y ) , q ( x, t, y ) i = f (Ω(1 − t )) f (Ω) h x, x i + f (Ω t ) f (Ω) h y, y i + 2 f (Ω(1 − t )) f (Ω t ) f (Ω) h x, y i = − α f (Ω) ( f (Ω(1 − t )) + f (Ω t ) + 2 f (Ω(1 − t )) f (Ω t ) g (Ω))= − α f (Ω) ( f (Ω − Ω t ))( f (Ω) g (Ω t ) + f (Ω t ) g (Ω)) + f (Ω t ))= − α f (Ω) ( f (Ω) g (Ω t ) + f (Ω t )(1 − g (Ω))= − α ( g (Ω t ) − α f (Ω t )) = − α . Now, as X is a subspace containing x and y and q ( x, t, y ) is a linearcombination of x and y , we conclude that it is also in X . Axioms (X1) , (X2) , (X3) of a mobi space are a direct consequence of the definitionof q . To prove (X4) we will use the notation Ω ′ ≡ θ ( x, y ′ ). If x = y and x = y ′ , we have that q (cid:0) x, , y (cid:1) = q (cid:0) x, , y ′ (cid:1) implies f (cid:0) Ω2 (cid:1) f (Ω) ( x + y ) = f (cid:0) Ω ′ (cid:1) f (Ω ′ ) ( x + y ′ ) (46)Applying the inner product with x in both sides of this equation andusing properties (39) to (41) in the form f (Ω) = 2 f (cid:18) Ω2 (cid:19) g (cid:18) Ω2 (cid:19) and 1 + g (Ω) = 2 g (cid:18) Ω2 (cid:19) , we get 12 g (cid:0) Ω2 (cid:1) h x, x + y i = 12 g (cid:0) Ω ′ (cid:1) h x, x + y ′ i⇒ g (cid:0) Ω2 (cid:1) ( − α − α g (Ω)) = 12 g (cid:0) Ω ′ (cid:1) ( − α − α g (Ω ′ )) ⇒ g (cid:18) Ω2 (cid:19) = g (cid:18) Ω ′ (cid:19) = 0 . Going back to (46) with this result, we conclude y = y ′ . If x = y and x = y ′ , then q (cid:0) x, , y (cid:1) = q (cid:0) x, , y ′ (cid:1) . Indeed, q (cid:0) x, , x (cid:1) = q (cid:0) x, , y ′ (cid:1) would imply x = f (cid:16) Ω ′ (cid:17) f (Ω ′ ) ( x + y ′ ) ⇒ h x, x i = g ( Ω ′ ) h x, x + y ′ i⇒ g ( Ω ′ ) (1 + g (Ω ′ )) ⇒ g (cid:0) Ω ′ (cid:1) = 1 ⇒ y ′ = x, in contradiction with the hypothesis. The case x = y and x = y ′ issimilar. Obviously (X4) is also verified if x = y and x = y ′ . The proofof (X5) begin with the observation that: g [ θ ( q ( x, a, y ) , q ( x, c, y ))] = g [ θ ( x, y )( c − a )] . (47) Indeed, beginning with the left-hand side of (47), if y = x : − α h f (Ω(1 − a )) f (Ω) x + f (Ω a ) f (Ω) y, f (Ω(1 − c )) f (Ω) x + f (Ω c ) f (Ω) y i = f (Ω(1 − a )) f (Ω(1 − c )) f (Ω) + f (Ω a ) f (Ω c ) f (Ω)+ (cid:18) f (Ω(1 − a )) f (Ω c ) + f (Ω(1 − c )) f (Ω a ) f (Ω) (cid:19) g (Ω)= f (Ω) g (Ω a ) g (Ω c ) − g (Ω) f (Ω a ) f (Ω c ) + f (Ω a ) f (Ω c ) f (Ω)= g (Ω a ) g (Ω c ) − α f (Ω a ) f (Ω c )= g (Ω a − Ω c ) = g (Ω c − Ω a )If y = x , then g [ θ ( q ( x, a, y ) , q ( x, c, y ))] = − α h q ( x, a, y ) , q ( x, c, y ) i = − α h x, x i = 1 = g (0) = g [ θ ( x, y )( c − a )] . Now, because a, c ∈ [0 ,
1] and Ω ∈ I imply Ω | c − a | ∈ I , we canconclude, since g is injective in I, that | θ ( q ( x, a, y ) , q ( x, c, y )) | = | θ ( x, y )( c − a ) | = | Ω( c − a ) | . (48)By (42), this also imply that, for any b ∈ [0 , f [ θ ( q ( x, a, y ) , q ( x, c, y )) b ] f [ θ ( q ( x, a, y ) , q ( x, c, y ))] = f [Ω( c − a ) b ] f [Ω( c − a )] . With these results, we are able to prove (X5) . For simplification, weuse the notation ˆ c ≡ c − a . First, for q ( x, a, y ) = q ( x, c, y ) and x = y : q [ q ( x, a, y ) , b, q ( x, c, y )]= f [Ω ˆ c (1 − b )] f [Ω ˆ c ] q ( x, a, y ) + f [Ω ˆ c b ] f [Ω ˆ c ] q ( x, ˆ c + a, y )= f [Ω ˆ c − Ω ˆ c b ] f [Ω(1 − a )] + f [Ω ˆ c b ] f [Ω(1 − a ) − Ω ˆ c ] f [Ω ˆ c ] f (Ω) x + f [Ω ˆ c − Ω ˆ c b ] f [Ω a ] + f [Ω ˆ c b ] f [Ω a + Ω ˆ c ] f [Ω ˆ c ] f (Ω) y = g [Ω ˆ c b ] f [Ω(1 − a )] − f [Ω ˆ c b ] g [Ω(1 − a )] f (Ω) x + g [Ω ˆ c b ] f [Ω a ] + f [Ω ˆ c b ] g [Ω a ] f (Ω) y = f [Ω(1 − a − ˆ c b )] f (Ω) x + f [Ω( a + ˆ c b )] f (Ω) y = q [ x, a + ( c − a ) b, y ] . If q ( x, a, y ) = q ( x, c, y ), on the one hand q ( q ( x, a, y ) , b, q ( x, c, y )) = q ( x, a, y ). On the other hand, from (47), we conclude that x = y or c = a and in both cases q ( x, a + b ( c − a ) , y ) = q ( x, a, y ). (cid:3) Before going to Proposition 7.2 that will explain how we can still geta mobi space out of a
Slerp type formula on the n -sphere despite thefact that geodesics between antipodal points are not unique, let us takea closer look to the formula (45) in the case of S n . This formula justgives the intersection between S n and a plane that contains the originand the points x and y , when x and y are not collinear. Starting at x when t = 0, a particle that goes to y on that plane at constant speedwill be, at an instant t ∈ [0 ,
1] at q ( x, t, y ) = cos(Ω t ) x + sin(Ω t ) z (49)where cos(Ω) x + sin(Ω) z = y. (50)Because h x, x i E = h y, y i E = 1 and Ω ≡ θ ( x, y ) = arccos h x, y i E , wehave that h x, z i E = 0 and h z, z i E = 1. When sin(Ω) = 0, we can justsolve (50) to obtain z and then (49) reads as expected: q ( x, t, y ) = sin[Ω (1 − t )]sin(Ω) x + sin[Ω t ]sin(Ω) y. When Ω = 0, which means x = y , there is no journey to make: q ( x, t, x ) = x . When Ω = π , which means y = − x , we have to choosethe plane we wish to travel on. Equivalently, we have to chose thedirection v ( x ) ∈ R n +1 we want to be playing the role of z . Of course,we still need h x, v ( x ) i E = 0 and h v ( x ) , v ( x ) i E = 1. There is one more condition: to get a mobi space, we also need v to be an even map be-cause q ( x, t, − x ) = q ( − x, − t, x ) (property (Y1) ) which means thatin a round trip, the going and the return must be done on the samepath. Proposition 7.2.
Consider the euclidean n -sphere X = { x ∈ R n +1 | h x, x i = 1 } , a map v : X → X such that v ( − x ) = v ( x ) and h x, v ( x ) i = 0 , and the map θ : X × X → [0 , π ] defined by θ ( x, y ) = arccos( h x, y i ) . With q : X × [0 , × X → X defined by q ( x, t, y ) = ( sin[ θ ( x,y ) (1 − t )]sin[ θ ( x,y )] x + sin[ θ ( x,y ) t ]sin[ θ ( x,y )] y , if θ ( x, y ) ∈ ]0 , π [cos[ θ ( x, y ) t ] x + sin[ θ ( x, y ) t ] v ( x ) , if θ ( x, y ) ∈ { , π } , ( X, q ) is a mobi space over the canonical mobi algebra.Proof. Most of the proof is the same as the proof of Proposition 7.1.We just have to consider the extra case where y = − x correspondingto Ω ≡ θ ( x, y ) = π . When Ω = π , we have q ( x, , y ) = cos(0) x +sin(0) v ( x ) = x and q ( x, , y ) = cos( π ) x + sin( π ) v ( x ) = − x = y , soAxioms (X1) , (X2) and (X3) of a mobi space are verified. Regarding (X4) , when Ω = π and Ω ′ ∈ ]0 , π [, we have that q (cid:0) x, , y (cid:1) = q (cid:0) x, , y ′ (cid:1) .Indeed q (cid:0) x, , y (cid:1) = q (cid:0) x, , y ′ (cid:1) implies v ( x ) = f (cid:0) Ω ′ (cid:1) f (Ω ′ ) ( x + y ′ ) ⇒ g (cid:18) Ω ′ (cid:19) ⇒ Ω ′ = π, in contradiction with Ω ′ ∈ ]0 , π [. The case Ω = π and Ω ′ = 0 is alsoincompatible with q (cid:0) x, , y (cid:1) = q (cid:0) x, , y ′ (cid:1) because v ( x ) = x . Inter-changing y and y ′ in the previous situations gives similar results. Thecase Ω = π and Ω ′ = π implies y = − x = y ′ , therefore (X4) is verified.Regarding (X5) , we first observe that (47) is valid for all x, y ∈ X .Indeed, if y = − x , thencos[ θ ( q ( x, a, y ) , q ( x, c, y ))] = h q ( x, a, y ) , q ( x, c, y ) i = h cos( πa ) x + sin( πa ) v ( x ) , cos( πc ) x + sin( πc ) v ( x ) i = cos( πa ) cos( πc ) + sin( πa ) sin( πc )= cos[ π ( a − c )] . So, we have that, ∀ x, y ∈ X : θ ( q ( x, a, y ) , q ( x, c, y )) = θ ( x, y ) | c − a | . (51) From equation (51), we conclude that θ ( q ( x, a, y ) , q ( x, c, y )) = π if andonly if Ω = π and | c − a | = 1 and that θ ( q ( x, a, y ) , q ( x, c, y )) = 0 if andonly if Ω = 0 or c = a . Therefore, besides the cases already proved inProposition 7.1, we have to consider the following four situations:(1) θ ( q ( x, a, y ) , q ( x, c, y )) = π , Ω = π and(a) c = 0, a = 1(b) c = 1, a = 0(2) θ ( q ( x, a, y ) , q ( x, c, y )) = 0, Ω = π and c = a (3) θ ( q ( x, a, y ) , q ( x, c, y )) ∈ ]0 , π [, Ω = π , c = a and | c − a | 6 = 1.For the situation (1a): q [ q ( x, a, y ) , b, q ( x, c, y )] = q ( − x, b, x ) = cos( πb )( − x ) + sin( πb ) v ( − x ) q [ x, a + b ( c − a ) , y ] = q ( x, − b, − x )= cos( π − πb ) x + sin( π − πb ) v ( x )= − cos( πb ) x + sin( πb ) v ( x ) . The Axiom (X5) is ensured through the hypothesis v ( − x ) = v ( x ). Forthe situation (1b): q [ q ( x, a, y ) , b, q ( x, c, y )] = q ( x, b, − x ) = cos( πb ) x + sin( πb ) v ( x ) q [ x, a + b ( c − a ) , y ] = q ( x, b, − x ) = cos( πb ) x + sin( πb ) v ( x ) . In situation (2), c = a and (X3) implies (X5) . Using ˆ c ≡ c − a , wehave for the situation (3): q [ q ( x, a, y ) , b, q ( x, c, y )]= sin[ π ( c − a )(1 − b )]sin[ π ( c − a )] (cos( πa ) x + sin( πa ) v ( x ))+ sin[ π ( c − a ) b ]sin[ π ( c − a )] (cos( πc ) x + sin( πc ) v ( x ))= sin[ π ˆ c (1 − b )] cos( πa ) + sin[ π ˆ c b ] cos( π (ˆ c + a ))sin[ π ˆ c ] x + sin[ π ˆ c (1 − b )] sin( πa ) + sin[ π ˆ c b ] sin( π (ˆ c + a ))sin[ π ˆ c ] v ( x )= cos[ π ˆ c b ] cos( πa ) − sin[ π ˆ c b ] sin[ πa ] x + cos[ π ˆ c b ] sin( πa ) + sin[ π ˆ c b ] cos[ πa ] v ( x )= cos[ π ( a + ˆ c b )] x + sin[ π ( a + ˆ c b )] v ( x )= q ( x, a + b ( c − a ) , y ) (cid:3) To finish this section, we present three examples of the map v usedin Proposition 7.2. First, consider the 1-sphere i.e. the circle. Wecan choose to move between antipodal points in the anticlockwise di-rection when starting somewhere at the top of the circle and in the clockwise direction when starting at the bottom. More specifically, if x = (cos θ, sin θ ) , θ ∈ [0 , π [, then v is defined as v ( x ) = (cid:26) ( − sin θ, cos θ ) if θ ∈ [0 , π [(sin θ, − cos θ ) if θ ∈ [ π, π [ . Secondly, let us choose to connect two antipodal points on S , differentfrom the poles, through the north pole and link the poles (on the z-axis) through the positive x-axis. This gives the following choice for v ,considering x + x + x = 1: v ( x , x , x ) = ( − x x , − x x , − x ) p − x if x = ± , , if x = ± . As a third example, consider the 2-sphere parametrized in sphericalcoordinates as: { (sin ϕ cos θ, sin ϕ sin θ, cos ϕ ) , ( θ, ϕ ) ∈ ([0 , π [ × ]0 , π [) ∪ (0 , ∪ (0 , π ) } . A possible map v is the following: v (sin ϕ cos θ, sin ϕ sin θ, cos ϕ )= ( ( − sin θ, cos θ, if ϕ ∈ (cid:2) , π (cid:2) or (cid:0) ϕ = π , θ ∈ [0 , π [ (cid:1) (sin θ, − cos θ, if ϕ ∈ (cid:3) π , π (cid:3) or (cid:0) ϕ = π , θ ∈ [ π, π [ (cid:1) . In this example, v is on the equator in a plane rotated π around thez-axis from the meridian of x . The choice θ = 0 for the poles con-nects them through the positive y-axis. The other antipodal points areconnected through a path that stays between the parallels of the twopoints, with an arbitrary choice for antipodal points on the equator.8. Conclusion
We have introduced a new algebraic structure which captures somefeatures of geodesic paths. The particular affine case was studied. Theexample of geodesics on the n -sphere was deduced from the formulaSlerp (spherical linear interpolation). Other interesting lines of studyinclude the connection with affine geometry [1, 10, 11] or the geometryof geodesics [2, 3]. The presence of an operation x ⊕ y = q ( x, ⁄ , y ) ad-mitting cancellation, together with the property x ⊕ y = y ⊕ x , tells usthat the category of mobi spaces is a weakly Mal’tsev category [13, 12].This was in fact the starting point that originated our investigation onmobi spaces. Further studies on spaces in which geodesics are not nec-essarily unique (such as the sphere) are worthwhile pursuing to betterunderstand the importance of the choices that have to be made. Thisand other topics, such as the development of a homology theory formobi spaces, will be investigated in future work. Acknowledgement
We thank Anders Kock for his helpful comments and suggestions toimprove the text.
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