Complex numbers form a two-dimensional commutative associative algebra with an identity. Up to a suitable choice of a basis there are exactly three different types of such algebras—see [241]. They are spanned by a basis consisting of 1 and a hypercomplex unit ι. The square of ι is −1 for complex numbers, 0 for dual numbers and 1 for double numbers. In these cases, we write the hypercomplex unit ι as i, ε and є, respectively.
The arithmetic of hypercomplex numbers is defined by associative, commutative and distributive laws. For example, the product is
| (u+ι v)(u′+ι v′)=(uu′+ι2vv′)+ι(uv′+u′v), where ι2=−1, 0, or 1. |
Further comparison of hypercomplex numbers is presented in Fig. B.2.
Despite significant similarities, only complex numbers belong to mainstream mathematics. Among their obvious advantages is the following:
The first property is not very crucial, since zero divisors can be treated through appropriate techniques, e.g. projective coordinates, cf. Section 8.1. The property of being algebraically-closed was not used in the present work. Thus, the absence of these properties is not an insuperable obstacle in the study of hypercomplex numbers. On the other hand, hypercomplex numbers naturally appeared in Section 3.3 from SL2(ℝ) action on the three different types of homogeneous spaces.
We call cycles three types of curves: circles, parabolas and equilateral hyperbolas. They belong to a large class of conic sections, i.e. they are the intersection of a cone with a plane, see Fig. 1.3. Algebraically, cycles are defined by a quadratic equation (1) and are a subset of quadrics.
The beauty of conic sections has attracted mathematicians for several thousand years. There is an extensive literature—see [123]*§ 6 for an entry-level introduction and [36]*Ch. 17 for a comprehensive coverage. We list below the basic definitions only in order to clarify the distinction between the classical foci, the centres of conic sections and our usage.
Figure B.2: The correspondence between complex, dual and double numbers.
We use the notation | P1P2 | and | P l | for the Euclidean distance between points P1, P2 and between a point P and a line l.
The above definition in terms of distances allows us to deduce the equality of the respective angles in each case—see Fig. B.1 and [123]*§ 6. This implies reflection of the respective rays. For example, any ray perpendicular to the directrix is reflected by the parabola to pass its focus—the “burning point”. There are many applications of this, from the legendary burning of the Roman fleet by Archimedes to practical (parabolic) satellite dishes.
The profound book by Yaglom [339] is already a golden classic appreciated by several generations of mathematicians and physicists. To avoid confusion, we provide a comparison of our notions and results with Yaglom’s.
Firstly, there is a methodological difference. Yaglom started from notions of length and angles and then derived objects (notably parabolas) which carry them out in an invariant way. We worked in the opposite direction by taking invariant objects (FSCc matrices) and deriving the respective notions and properties, which were also invariant. This leads to significant distinctions in our results which are collected in Fig. B.3.
90
Notion Yaglom’s usage This work Circle Defined as a locus of equidistant points in metric d(u,v; u′,v′)=| u−u′ |. Effectively is a pair of vertical lines. A limiting case of p-cycles with n=0. Forms a Möbius-invariant subfamily of selfadjoint p-cycles (Definition 12). In this case, all three centres coincide. We use term “circle” only to describe a drawing of a cycle in the elliptic point space ℝe. Cycle Defined as locus of points having fixed angle view to a segment. Effectively is a non-degenerate parabola with a vertical axis. We use this word for a point of the projective cycle space ℙ3. Its drawing in various point spaces can be a circle, parabola, hyperbola, single or pair of lines, single point or an empty set. Centre Absent, Yaglom’s cycles are “centreless”. A cycle has three EPH centres. Diameter A quarter of the parabola’s focal length. The distance between real roots. Special lines Vertical lines, special role reflects absolute time in Galilean mechanics. The intersection of invariant sets of selfadjoint and zero radius p-cycles, i.e. having the form (1,l,0,l2). Orthogonal, perpendicular The relation between two lines, if one of them is special. Delivers the shortest distance. We have a variety of different orthogonality and perpendicularity relations, which are not necessary local and symmetric. Inversion in circles. Defined through the degenerated p-metric Conjugation with a degenerate parabola (n=0). Reflection in cycles Defined as a reflection in the parabola along the special lines. Composition of conjugation with three parabolas—see Exercise 33.
Figure B.3: Comparison with the Yaglom book
In short, we tried to avoid an overlap with Yaglom’s book [339]—our results are either new or obtained in a different manner.
We may also note some later attempts to axiomatize the theory, see for example [315]. As it often happens with axiomatisation, the result may be compared with sun-dried fruits: it is suitable for long-term preservation but does not keep all its test, aroma and vitamins. Also, the fruit needs to be grown up within the synthetic approach beforehand.
The development of parabolic and hyperbolic analogues of complex analysis has a long but sporadic history. In the absence of continuity, there are many examples when a researcher started from a scratch without any knowledge of previous works. There may be even more forgotten papers on the subject. To improve the situation, we list here some papers without trying to be comprehensive or even representative.
The survey and history of Cayley–Klein geometries is presented in Yaglom’s book [339] and this will be completed by the work [284] which provides the full axiomatic classification of EPH cases. See also [311, 270, 304] for modern discussion of the Erlangen programme. A search for hyperbolic function theory was attempted several times starting from the 1930s—see, for example, [332, 241, 260, 253, 286]. Despite some important advances, the obtained hyperbolic theory is not yet as natural and complete as complex analysis is. Parabolic geometry was considered in [339] with rather trivial “parabolic calculus” described in [58]. There is also interest in this topic from different areas: differential geometry [342, 29, 58, 57, 102, 229, 230, 308, 44, 19, 79], kinematics [232, 331, 339, 78], quantum mechanics [142, 143, 120, 118, 151, 154, 155, 157, 9, 133, 325, 327], group representations [115, 116], space-time geometry [42, 141, 130, 131, 256], hypercomplex analysis [170] [60] [90] [91] [243] [234] [235]*Part IV and non-linear dynamics [281, 282, 283], integrability of discrete equations [41, 226, 227, 228, 297].
Our CAS uses FSCc, which is expressed through Clifford algebras [191, 211]. Clifford algebras also admit straightforward generalisation to higher dimensions. There are two slightly different forms: so called vector (all versions of CAS) and paravector (since v.3.0 only) formalisms. In paravector formalism a point with coordinates(u,v) is represented by the paravector u1+ve0 from the Clifford algebra with the single generator e0 and unit 1. Such Clifford algebra is commutative and effectively isomorphic to complex, dual or double numbers depending on the value e02. Therefore, all FSCc matrix formulae for the paravector formalism coincide with ones presented in this book.
In vector formalism a point with coordinates (u,v) is represented by the vector ue0+ve1 from the Clifford algebra with two generators e0 and e1 which anticommute: e0e1=−e1e0. Although, in this case, we need to take care on the non-commutativity of Clifford numbers, many matrix expressions have a simpler form. We briefly outline differences in FSCc for vector formalism below.
Let Cl(σ) be the four-dimensional Clifford algebra with unit 1, two generators e0 and e1, and the fourth element of the basis, their product e0e1. The multiplication table is e02=−1, e12=σ and e0e1=−e1e0. Here, σ=−1, 0 and 1 in the respective EPH cases. The point space ℝσ consists of vectors u e0+ ve1. An isomorphic realisation of SL2(ℝ) is obtained if we replace a matrix (
| a | b |
| c | d |
) by (
| a | b e0 |
| −c e0 | d |
) for any σ. The Möbius transformation of ℝσ→ ℝσ for all three algebras Cl(σ) by the same expression, cf. (4), is
| : ue0+ve1 ↦ |
| , (1) |
where the expression a/b in a non-commutative algebra is understood as ab−1.
In Clifford algebras, the FSCc matrix of a cycle (k,l,n,m) is, cf. (5),
| Cσcs = |
| , with ĕ02=−1, ĕ12=σc, (2) |
where the EPH type of Cl(σc) may be different from the type of Cl(σ). In terms of matrices (1) and (2) the SL2(ℝ)-similarity (7) has exactly the same form S σcs= gCσcsg−1. However, the cycle similarity (10) becomes simpler, e.g. there is no need in conjugation:
| Cσcs: S σcs ↦ CσcsS σcsCσcs. (3) |
The cycle product is ℜtr (CσcsS σcs) [191]. Its “imaginary” part (vanishing for hypercomplex numbers) is equal to the symplectic form of cycles’ centres.
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