Cycles Cross Ratio: an Invitation

In this chapter we review the cycles cross ratio introduced in [217]. This is an invariant with applications in Möbius and Lie sphere geometry [59] [32]*Ch. 3 [148] which is analogous to the cross-ratio of four points in projective [255] [303]*§ 9.3 [339]*App. A and hyperbolic geometries [24]*§ 4.4 [276]*III.5 [300]*§ I.5. To avoid technicalities and stay visual we will work in two dimensions, luckily all main features are already illuminated in this case. Our purpose is not to give an exhausting presentation (in fact, we are hoping it is far from being possible now), but rather to draw attention to the new invariant and its benefits to the various geometrical settings. There are several far-reaching generalisations in higher dimensions and non-Euclidean metrics, see Sect. 12.4 for a discussion. In abstract framework the new invariant can be considered in any projective space with a bilinear pairing.

12.1 Preliminaries: Projective space of cycles

We introduce our topic following its historic development. This conforms to our hands-on approach as an opposition to a start from the most general abstract construction.

Circles are the traditional subject of geometrical studies and their numerous properties were already widely presented in Euclid’s Elements. However, advances in their analytical presentations is not a so distant history.

by a point (g,f,c) in some subset of the three-dimensional Euclidean space ℝ³ was used in [276]*Ch. II. Abstractly, we can treat a point (x₀,y₀) of a plane as the zero radius circle with coefficients (g,f,c)=(−x₀,−y₀,x₀²+y₀²).

which also includes straight lines for k=0. This extension comes at a price: parameters (k,l,n,m) shall be treated as elements of the three dimensional projective space Pℝ³ rather than the Euclidean space ℝ⁴ since equations (2) with (k,l,n,m) and (k₁,l₁,n₁,m₁)= (λ k, λ l, λ n,λ m) defines the same set of points for any λ≠ 0. This parametrisation is known as tetracyclic/polyspheric coordinates, cf. [150]*§ 2.4.1 [36]*§ 20.7.

The next observation is that the linear structure of Pℝ³ is relevant for circles geometry. For example, the traditional concept of pencil of circles [71]*§ 2.3 is nothing else but the linear span in Pℝ³ [300]*§ I.1.c. Therefore, it will be convenient to accept all points (k,l,n,m)∈ Pℝ³ on equal ground even if they correspond to an empty set of solutions (x,y) in (2). The latter can be thought as “circles with imaginary radii”.

It is also appropriate to consider (0,0,0,1) as a representative of the point C_∞ at infinity, which complements ℝ² to the Riemann sphere. Following [339], we call circles (with real and imaginary radii), straight lines and points (including C_∞) on a plane by joint name cycles. Correspondingly, the space Pℝ³ representing them—the cycles space.

Sometimes, we need to consider cycles with an orientation. This can be used, for example, to distinguish “inner” and “outer” tangency of cycles [101, 215, 211]. Furthermore, oriented cycles and their tangency relations are the starting point for the Lie sphere geometry. It is easy to encode cycles’ orientations through polyspheric coordinates: cycles (k,l,n,m) and (k′,l′,n′,m′) have the opposite orientations if (k,l,n,m) = (λ k′, λ l′, λ n′, λ m′) for some negative λ and two points (k,l,n,m) and (k′,l′,n′,m′) are considered as representatives of the same oriented cycle if (k,l,n,m) = (λ k′, λ l′, λ n′, λ m′) for a positive λ .

An algebraic consideration often benefits from the introduction of complex numbers. For example, we can re-write (2) as:

where z=x+i y and L=l+i n. We call Fillmore–Springer–Cnops construction (or FSCc for short) the association of the matrix C= (

A reader may expect a more straightforward realisation of the quadratic form (2), cf. [300]*§ I.1 [306]*Thm. 9.2.11:

However, FSCc will show its benefits in the next section. Meanwhile, we note that a point z corresponding to the zero radius cycle with the centre z is represented by the matrix

where detZ=0. Also, the point at infinity can be represented by a zero radius cycle:

12.2 Fractional linear transformations and the invariant product

In the spirit of the Erlangen programme of Felix Klein (greatly influenced by Sophus Lie), a consideration of cycle geometry is based on a group of transformations preserving this family [185, 198]. Let M= (

) be an invertible 2× 2 complex matrix. Then the fractional linear transformation (FLT for short) of the extended complex plane $ℂ^_$= ℂ∪ {∞} is defined by:

) with complex conjugated entries of M. For a cycle C the matrix C corresponds to the reflection of C in the real axis y=0. Also, due to special structure of FSCc matrix we easily check that

If a cycle C is composed from a non-empty set of points in (x,y) ∈ ℝ² satisfying (2), then their images under transformation (7) form again a cycle C₁ with notable link to FSCc:

Lemma 1 Transformation (7) maps a cycle C= (

−m

−L

) into a cycle C₁= (

₁

−m₁

k₁

−L₁

) such that C₁= M C M⁻¹.

Clearly, the map C → M C M⁻¹ is meaningful for any cycle, including imaginary ones, thus we regard it as FLT action on the cycle space. For the matrix form (4) the above identity C₁= M C M⁻¹ needs to be replaced by the matrix congruence C₁= M^* C M [300]*§ II.6.e [306]*Thm. 9.2.13. This difference is significant in view of the following definition.

Definition 2 For two cycles C and C₁ define the cycles product by:

⟨ C,C₁ ⟩ = −tr(C

₁)   , (9)

where tr denotes the trace of a matrix.

We call two cycles C and C₁ orthogonal if ⟨ C,C₁ ⟩=0.

and observe that it is linear in coefficients of the cycle C (and C₁ as well). On the other hand, it is the initial definition (9), which allows us to use the invariance of trace under matrix similarity to conclude the following.

Corollary 3 The cycles product is invariant under the transformation C↦ M C M⁻¹. Therefore FLT (7) preserves orthogonality of cycles.

The cycle product is a rather recent addition to the cycle geometry, see independent works [100] [64] [65]*§ 4.1 [163]*§ 4.2. Interestingly, expression (9) essentially repeats the GNS-construction in C^*-algebras [14] which is older by half of century at least.

Example 4 [198]*Ch. 6 [212] [211] The cycles product and cycles orthogonality encode a great amount of geometrical characteristics. For example, for cycles represented by non-empty sets of points in ℝ² we note the following:

A cycle is a straight line if it is orthogonal ⟨ C,C_∞ ⟩=0 to the zero radius cycle at infinity C_∞ (6).
A cycle Z represents a point if Z is self-orthogonal (isotropic): ⟨ Z,Z ⟩=0. More generally, (8) implies:
⟨ C,C ⟩=2det(C) . (11)
A cycle C passes a point Z if they are orthogonal ⟨ C,Z ⟩=0.
A cycle C represents a line in Lobachevsky geometry[303]*Ch. 12 if it is orthogonal ⟨ C,C_ℝ ⟩=0 to the real line cycle C_ℝ= (
i 0

0 i

) .
Two cycles are orthogonal as subsets of a plane (i.e. they have perpendicular tangents at an intersection point) if they are orthogonal in the sense of Defn. 11.
Two cycles C and C₁ are tangent (i.e. have a unique point in common) if
⟨ C,C₁ ⟩² = ⟨ C,C ⟩ ⟨ C₁,C₁ ⟩ .
Inversive distance [71]*§ 5.8 θ of two (non-isotropic) cycles is defined by the formula:
θ =
⟨ C,C₁ ⟩

√

⟨ C,C ⟩ ⟨ C₁,C₁ ⟩

. (12)

In particular, the above discussed orthogonality corresponds to θ=0 and the tangency to θ=±1. For intersecting cycles θ is the cosine of the intersecting angle.
A generalisation of Steiner power d(C,C₁) of two cycles is defined as, cf. [101]*§ 1.1:
d(C,C₁)= ⟨ C,C₁ ⟩ + √

⟨ C,C ⟩ ⟨ C₁,C₁ ⟩

 , (13)

where both cycles C and C₁ are scaled to have k=1 and k₁=1. Geometrically, the generalised Steiner power for spheres provides the square of tangential distance.

Remark 5 The cycles product is indefinite, see [113] for an account of the theory with some refreshing differences to the more familiar situation of inner product spaces. One illustration is the presence of self-orthogonal non-zero vectors, see Ex. 2 above. Another noteworthy observation is that the product (10) has the Lorentzian signature (1,3) and ℝ⁴ with this product is isomorphic to Minkowski space-time [163]*§ 4.2.

12.3 Cycles cross ratio

Due to the projective nature of the cycles space (i.e. matrices C and λ C correspond to the same cycle) a non-zero value of the cycle product (9) is not directly meaningful. Of course, this does not affect the cycles orthogonality. A partial remedy in other cases is possible through various normalisations [198]*§ 5.2. Usually they are specified by either of the following conditions

Recall, that the projective ambiguity is elegantly balanced in the cross ratio of four points [24]*§ 4.4 [276]*III.5 [300]*§ I.5:

Definition 6 A cycles cross ratio of four cycles C₁, C₂, C₃ and C₄ is:

C₁, C₂; C₃, C₄=

⟨ C₁,C₃ ⟩

⟨ C₁,C₄ ⟩

⟨ C₂,C₃ ⟩

⟨ C₂,C₄ ⟩

(15)

assuming ⟨ C₁,C₄ ⟩ ⟨ C₂,C₃ ⟩ ≠ 0. If ⟨ C₁,C₄ ⟩ ⟨ C₂,C₃ ⟩ = 0 but ⟨ C₁,C₃ ⟩ ⟨ C₂,C₄ ⟩ ≠ 0 we put C₁, C₂; C₃, C₄ = ∞. The cycles cross ratio is generally undefined in the remaining case of an indeterminacy 0/0.

Note that some additional geometrical reasons may help to resolve the last situation, see the consideration of orthogonality/tangency with zero radius cycle in Ex. 9.

Proposition 7

The cycles cross ratio is a well-defined FLT-invariant of quadruples of cycles.
The cycles cross ratio of four zero radius cycles is the squared modulus of the cross ratio for the respective points:
Z₁, Z₂; Z₃, Z₄= ⎪
⎪ (z₁,z₂;z₃,z₄) ⎪
⎪ ². (16)
There is the cancellation formula:
C₁, C; C₃, C₄ C, C₂; C₃, C₄ = C₁, C₂; C₃, C₄   . (17)

Proof. The first statement follows from Cor. 3 and the construction of the cycles cross ratio. To show the second statement we derive from (5) and (10) that:

if we use representations of zero radius cycles Z_i and Z_j by coefficients with k_i=k_j=1. This implies (16). A demonstration of (17) is straightforward.

To demonstrate that there is more than just a formal similarity between the two we briefly list some applications. First, we rephrase Ex. 7 in new terms.

Example 8 The capacitance cap(C,C₁) of two cycles [144]*§ 5.1 [145]*Defn. 3 coincides with the following cycles cross product:

cap(C,C₁) = C,C₁;C₁,C = θ²,

where θ is the inversive distance (12). Thereafter, FLT-invariance of the cycles cross ratio implies that the intersection angle of cycles is FLT-invariant. In particular cycles are

orthogonal if C,C₁;C₁,C=0;
tangent if C,C₁;C₁,C=± 1; and
disjoint if | C,C₁;C₁,C |> 1.

Relation 1 is merely a consequence of the first-order orthogonality relation ⟨ C,C₁ ⟩=0, which is fundamental to conformal and incidence geometries, cf. Ex. 4(i)–(v). Meanwhile, the tangency condition (ii) is genuinely quadratic and shall be equally significant in Lie spheres geometry, the Steiner’s porism [144, 145], and other questions formulated purely in terms of cycles’ tangency.

Example 9 If a non-zero radius cycle passes a zero radius cycle (point) their cross ratio has an indeterminacy 0/0. Geometrically their relation can be seen in either ways: as orthogonality or tangency. Therefore, the indeterminacy of the cycle cross ratio can be geometrically resolved differently either to 0 (indicates orthogonality) or 1 (corresponds to tangency). More specifically (see the supporting symbolic computations in the notebook [216]):

Orthogonality. Consider a cycle Z_t with a fixed centre and a variable squared radius t. Take a generic cycle C orthogonal to Z_t. To resolve an indeterminacy 0/0 we use l’Hospital’s rule at the point t=0, which corresponds to Z_t becoming a zero radius cycle. This produces C,Z_t;Z_t,C|_t=0=0.
Tangency. For a zero radius cycle Z and passing it cycle C, consider a generic cycle C_t=(1−t)· Z+ t· C, t∈[0,1] from the pencil (linearly) spanned by Z and C. Since C_t touches C Ex. 2 implies that C_t, C; C, C_t=1 for t > 0 and C_t coincides with Z for t=0. Thus, we can extend the value 1 to [Z, C; C, Z] by continuity.

The last technique makes the cycles cross ratio meaningful for Lie spheres geometry, which extends FLT by non-point Lie transformations, when a non-zero radius cycle is sent to a zero radius one.

Example 10 The Steiner power (13) can be written as:

d(C,C₁)= C,C_ℝ;C₁,C_ℝ +

√

C,C_ℝ;C,C_ℝ

√

C₁,C_ℝ;C₁,C_ℝ

  , (18)

where C_ℝ is the real line and cycles C and C₁ do not need to be normalised in any particular way. Thereafter, the Steiner power is an invariant of two cycles C and C₁ under Möbius transformations, since they fix the real line C_ℝ. The Möbius invariance is not so obvious from expression (13).

The next two applications will generalise the main features of the traditional cross ratio. Recall the other name of the cross ratio—the anharmonic ratio. The origin of the latter is as follows. Two points z₁ and z₂ on a line define a one-dimensional sphere with the centre O=1/2(z₁+z₂), which can be taken as the origin. Two points c₁ and c₂ are called harmonically conjugated (with respect to z₁ and z₂) if:

Thus, the cross ratio can be viewed as a measure how far four points are from harmonic conjugation, i.e. a measure of anharmonicity of a quadruple.

To make a similar interpretation of the cycles cross ratio recall that for FSCc matrices of a cycle C₁ and its reflection C₂ in a cycle C we have: C₂=CC₁C, cf. [198]*§ 6.5. That is, the reflection in a cycle C is the composition of FLT transform with FSCc matrix C and complex conjugation of matrix entries. It is easy to obtain the following:

Proposition 11 If a cycle C₁ is a reflection of C₂ in a cycle C then:

C₁, C; C, C₁ = C₂, C; C, C₂.

More generally, the reflection in a cycle preserves the inversive distance, cf. Ex. 8.

The above condition is necessary, we describe a sufficient one as a figure in the sense of [212] [211] [215]. In short, a figure is an ensemble of cycles interrelated by cycles’ relations. For the purpose of this chapter an FLT-invariant relation “to be orthogonal” between two cycles is enough. Software implementations of these figures can be found in [216].

Construction 12

For two given cycles C and C₁ construct the reflection C₂=CC₁C of C₁ in C.
Take any cycle C_o orthogonal to C and C₁, see Ill. 12.1. All such cycles make a pencil—one dimensional subspace of the projective space of cycles. That is because there are only two linear equations for orthogonality (10) to determine four projective coordinates (k_o,l_o,n_o,m_o). By Prop. 11 C_o is also orthogonal to C₂.
Define cycles by orthogonality to C, C_o and itself (zero radius condition), that is the intersection points of C and C_o. Since self-orthogonality is a quadratic condition there are two solutions: Z₁ and Z₂.
The harmonic conjugation of C₁ and C₂ (their reflection in C) implies, cf. (19):
C₁,C₂;Z₁,Z₂=1 . (20)

This is demonstrated by symbolic computation in [216].

As the final illustration, we introduce Möbius-invariant distance between cycles. Recall that FLT with an SL₂(ℝ) matrix fixes the real line and it is called a Möbius transformation [185] [198]*Ch. 1 [306]*§ 9.3. The corresponding figure is as follows, see Ill. 12.2:

Construction 13

Let two distinct cycles C₁ and C₂ be given and they are different from the real line C_ℝ.
Define a cycle C to be orthogonal to C₁, C₂, and C_ℝ. It is specified by three linear equations for homogeneous coordinates (k, l, n, m). In generic position a solution is unique, however it can be an imaginary cycle (with a negative square of the radius).
Define cycles by orthogonality to C, C_ℝ and itself, that is the intersection points of C and C_ℝ. In general position there are two solutions which we denote by Z₁ and Z₂. For an imaginary cycle C first coordinates of Z₁ and Z₂ are conjugated complex numbers.
Since the entire construction is completely determined by the given cycles C₁ and C₂ we define the distance between two cycles by:
d(C₁,C₂)=
1

2

logC₁,C₂; Z₁,Z₂. (21)

From Möbius invariance of the real line and cycles cross ratio our construction implies the following:

Proposition 14

Distance (21) is Möbius invariant;
For zero radius cycles C₁, C₂ formula (21) coincides with the Lobachevsky metric on the upper-half plane.
For any cycle C₃ orthogonal to C the (signed) distance is additive: d(C₁,C₃)=d(C₁,C₂)+d(C₂,C₃).
If centres of C₁ and C₂ are on the imaginary axis (therefore Z₁ and Z₂ are zero and infinity) then d(C₁,C₂)=log(m₁/k₁)−log(k₂/m₂).

Obviously, Prop. 2 is the consequence of (16) and 3 follows from the cancellation rule (17). On the other hand, the expression in 4 can be obtained by a direct computation, see [216]. Note, that in this case m_i/k_i is the square tangential distance (also known as the generalised Steiner power (13), (18)) from Z₁ (the origin) to the cycle C_i. Thus, for a zero radius C_i it coincides with the usual distance between centres of Z₁ and C_i and Prop. 4 recreates the classical result [24]*(7.2.6).

Comparing constructions on Ill. 12.1–12.2 and formulae (20)–(21) we can say that the invariant distance measures how far two cycles are from being reflections of each other in the real line.

There is an alternative way to define a distance between spheres using cross ratio. Recall the formula [26]*(15.2.1)

The alternative distance is defined again via the cross ratio although of the different form:

Note, that this time we do not use additional cycles besides the Möbius-invariant real line.

The connection between the definition (21) of the distance and the new one is given by the formula

which are vertically aligned. Without loss of generality this can be always achieved by a suitable Möbius transformation. Another technical assumption is that cycles do not intersect the real axis, which simplifies the symbolic computations. With all of these we get:

There is the clear inconsistency of two expressions by the presence of parameters v and v₁ in the second one. Yet, both formulae give the same distances for points, which correspond to the case d=v and d₁=v₁ in (23). Obviously, we get the value sinh²1/2d_a(C, C₁) = 1/4 (v−v₁)²/vv₁ agreeing with (22).

Further example with geometric significance is the cycle distance from a point (0,v) to a cycle (1,0, v₁, v₁²−r²) with a radius r. The answer is

That is the tangential distance (v−v₁)²−r² to the cycle modified by the same "hyperbolic factor" 1/(vv₁). as in the point case (22).

12.4 Discussion and generalisations

We presented some evidence that the cycles cross ratio extends to Lie spheres geometry the concept of the cross ratio of four points. It is natural to expect that a majority of the classic theory [255] [303]*§ 9.3 [339]*App. A [24]*§ 4.4 [276]*III.5 [300]*§ I.5 admits similar adaptation as well. However, we can expect even more than that.

One can lay down a general framework for the introduced invariant (15) in generic projective spaces as follows¹. Let V be a vector space over an arbitrary field F with a bilinear pairing ⟨ ·,· ⟩: V × V → F. Upon choosing any four points of the projective space on V one may select arbitrary non-zero vectors, say c₁, c₂, c₃, c₄, representing these points. Then the quotient

will in general be an element of F. Whenever this scalar is well defined it obviously will be an invariant under the natural action of the general orthogonal group GO(V, ⟨ ·,· ⟩) on the point set of the underlying projective space. More generally, a sort of projective space can be defined on a module V for a weaker structure than a field, say, algebras of dual and double numbers [261]. In such cases numerous divisors of zero prompt a projective treatment [52] [198]*§ 4.5 of the new cross ratio (25). Another perspective direction to research are discrete Möbius geometries [144, 145].

The geometric applications of the new invariant (15), (25) are expected much beyond the currently presented situation of circles on a plane. Indeed, FSCc and cycles product based on Clifford algebras works in spaces of higher dimensions and with non-degenerate metrics of arbitrary signatures [100] [64] [65]*§ 4.1 [198]. In a straightforward fashion cycles cross ratio (15) remains a geometric FLT-invariant in higher dimensions as well.

A more challenging situation occurs if we have a degenerate metric and cycles are represented by parabolas [339] and respective FLT are based on dual numbers [44]. Besides theoretical interest such spaces are meaningful physical models [339, 118, 117, 150]. The differential geometry loses its ground in the degenerate case and non-commutative/non-local effects appear [198]*§ 7.2. The presence of zero divisors among dual numbers prompts a projective approach [52] [198]*§ 4.5 to the cross ratio of four dual numbers, which replaces (14). Variational methods do not produce FLT-invariant family of geodesics in the degenerate case, instead geometry of cycles needs to be employed [165]. Our construction from Figure 13 shall be usable to define Möbius invariant distance between parabolic cycles as well, cf. [198]*§ 9.5.

Last but not least, if we restrict the group of transformations from FLT to Möbius maps (or any other subgroup of FLT which fixes a particular cycle) we will get a larger set of invariants. In particular, the FSCc matrix (3) and the corresponding cycles product (9) can use different number systems (complex, dual or double) independently from the geometry of the plane [185] [198]*§ 5.3 [209]. Therefore, there will be three different cycles cross ratios (15) for each geometry of circles, parabolas and hyperbolas. This echoes the existence of nine Cayley–Klein geometries [339]*App. A [115] [284] .

Various aspects of the cycles cross ratio appear to be a wide and fruitful field for further research.

I am grateful to an anonymous referee for pointing this out in his/her report.

α	β
γ	δ

Lecture 12 Cycles Cross Ratio: an Invitation

12.1 Preliminaries: Projective space of cycles

12.2 Fractional linear transformations and the invariant product

12.3 Cycles cross ratio

12.4 Discussion and generalisations