Joint Invariants of Cycles: Orthogonality

The invariant cycle product, defined in the previous chapter, allows us to define joint invariants of two or more cycles. Being initially defined in an algebraic fashion, they also reveal their rich geometrical content. We will also see that FSCc matrices define reflections and inversions in cycles, which extend Möbius maps.

6.1 Orthogonality of Cycles

According to the categorical viewpoint , the internal properties of objects are of minor importance in comparison with their relations to other objects from the same class. Such a projection of internal properties into external relations was also discussed at the beginning of Section 4.2. As a further illustration, we may give the proof of Theorem 13, outlined below. Thus, we will now look for invariant relations between two or more cycles.

After we defined the invariant cycle product (3), the next standard move is to use the analogy with Euclidean and Hilbert spaces and give the following definition:

Definition 1 Two cycles C_{σ_c}^s and S _{σ_c}^s are called σ_c-orthogonal if their σ_c-cycle product vanishes:

⟨ C_{σ_c}^s,S _{σ_c}^s ⟩=0. (1)

Exercise 2 Use Exercise 12 to check the following:

The σ_c-orthogonality condition (1) is invariant under Möbius transformations.
The explicit expression for σ_c-orthogonality of cycles in terms of their coefficients is
2σ_cn′n−2l′l+k′m+m′k=0. (2)
The σ_c-orthogonality of cycles defined by their e-centres (u,v) and (u′, v′) with σ-determinants −r² and −r′², respectively, is:
(u−u′)²−σ(v−v′)²−2(σ−σ_c)vv′−r²−r′²=0. (3)
Two circles are e-orthogonal if their tangents at an intersection point form a right angle. Hint: Use the previous formula (3), the inverse of Pythagoras’ theorem and Fig. 6.1(a) for this. ⋄

The last item can be reformulated as follows: For circles, their e-orthogonality as vectors in the cycle spaces ℙ³ with the cycle product (3) coincides with their orthogonality as geometrical sets in the point space ℝ_e. This is very strong support for FSCc and the cycle product (3) defined from it. Thereafter, it is tempting to find similar interpretations for other types of orthogonality. The next exercise performs the first step for the case of σ-orthogonality in the matching point space ℝ_σ.

Exercise 3 Check the following geometrical meaning for σ-orthogonality of σ-cycles.

(e,h)

Let σ=± 1. Then, two cycles in ℝ_σ (that is, circles or hyperbolas) are σ-orthogonal if slopes S₁ and S₂ of their tangents at the intersection point satisfy the condition

S₁ S₂=σ. (4)

The geometrical meaning of this condition can be given either in terms of angles (A) or centres (C):

For the case σ=−1 (circles), equation (4) implies orthogonality of the tangents, cf. Exercise 4. For σ=1, two hyperbolas are h-orthogonal if lines with the slopes ± 1 bisect the angle of intersection of the hyperbolas, see Fig. 6.1(b). Hint: Define a cycle C_σ by the condition that it passes a point (u,v)∈ℝ_σ. Define a second cycle S _σ by both conditions: it passes the same point (u,v) and is orthogonal to C_σ. Then use the implicit derivative formula to find the slopes of tangents to C_σ and S _σ at (u,v). A script calculating this in CAS is also provided. ⋄
In the cases σ=± 1, the tangent to one cycle at the intersection point passes the centre of another cycle. Hint: This fact is clear for circles from inspection of, say, Fig. 6.1(a). For hyperbolas, it is enough to observe that the slope of the tangent to a hyperbola y=1/x at a point (x, 1/x) is −1/x² and the slope of the line from the centre (0,0) to the point (x,1/x) is 1/x², so the angle between two lines is bisected by a vertical/horizontal line. All our hyperbolas are obtained from y=1/x by rotation of ± 45 ^∘ and scaling. ⋄

(p)

Let σ=0 and a parabola C_p have two real roots u₁ and u₂. If a parabola S _p is p-orthogonal to C_p, then the tangent to S _p at a point above one of the roots u_1,2 passes the p-centre (u₁+u₂/2,0) of C_p.

Remark 4 Note that the geometric p-orthogonality condition for parabolas is non-local in the sense that it does not direct behaviour of tangents at the intersection points. Moreover, orthogonal parabolas need not intersect at all. We shall see more examples of such non-locality later. The relation 3(p) is also another example of boundary awareness, cf. Remark 3 —we are taking a tangent of one parabola above the point of intersection of the other parabola with the boundary of the upper half-plane.

The stated geometrical conditions for orthogonality of cycles are not only necessary but are sufficient as well.

Exercise 5

Prove the converses of the two statements in Exercise 3. Hint: To avoid irrationalities in the parabolic case and make the calculations accessible for CAS, you may proceed as follows. Define a generic parabola passing (u,v) and use implicit derivation to find its tangent at this point. Define the second parabola passing (u,0) and its centre at the intersection of the tangent of the first parabola at (u,v) and the horizontal axis. Then, check the p-orthogonality of the two parabolas. ⋄
Let a parabola have two tangents touching it at (u₁,v₁) and (u₂,v₂) and these tangents intersect at a point (u,v). Then, u=u₁+u₂/2. Hint: Use the geometric description of p-orthogonality and note that the two roots of a parabola are interchangeable in the necessary condition for p-orthogonality. ⋄

We found geometrically necessary and sufficient conditions for σ-orthogonality in the matching point space ℝ_σ. The remaining six non-matching cases will be reduced to this in Section 6.3 using an auxiliary ghost cycle. However, it will be useful to study some more properties of orthogonality.

6.2 Orthogonality Miscellanea

The explicit formulae (2) and (3) allow us to obtain several simple and yet useful conclusions.

Exercise 6 Show that:

A cycle is σ_c-self-orthogonal (isotropic) if and only if it is σ_c-zero-radius cycle (11).
For σ_c=± 1, there is no non-trivial cycle orthogonal to all other non-trivial cycles. For σ_c=0, only the horizontal axis v=0 is orthogonal to all other non-trivial cycles.
If two real (e-negative) circles are e-orthogonal, then they intersect. Give an example of h-orthogonal hyperbolas which do not intersect in ℝ_h. Hint: Use properties of the Cauchy–Schwarz inequality from Exercise 1. ⋄
A cycle C_σ^s is σ-orthogonal to a zero-radius cycle Z_σ^s (11) if and only if σ-implementation of C_σ^s passes through the σ-centre of Z_σ^s, or, analytically,
k(u² − σ v²) − 2⟨ (l,n),(u, σ_c v) ⟩+m =0. (5)
For σ_c=± 1, any cycle is uniquely defined by the family of cycles orthogonal to it, i.e. (C_{σ_c}^s^⊥)^⊥={C_{σ_c}^s}.
For σ_c= 0, the set (C_{σ_c}^s^⊥)^⊥ is the pencil spanned by C_{σ_c}^s and the real line. In particular, if C_{σ_c}^s has real roots, then all cycles in (C_{σ_c}^s^⊥)^⊥ have these roots.
Two σ_c-zero-radius cycles are σ_c-orthogonal if:
1. for σ_c=−1 and σ=±1, cycles’ σ-centres coincide.
2. for σ_c=0 and σ=±1, cycles’ σ-centres belong to the same vertical line.
3. for σ_c=1 and σ=±1, each cycle’s σ-centre belongs to the light cone defined by the other cycle.
(Note, that in the Benz axiomatisation [315]*§ 1 these relations between cycles’ centres are called parallelism .)
Hint: Use the explicit expression for the cycle product (12). ⋄

The connection between orthogonality and incidence from Exercise 4 allows us to combine the techniques of zero-radius cycles and orthogonality in an efficient tool.

Exercise 7 Fill all gaps in the following proof:

Proof.[Sketch of an alternative proof of Theorem 13] We already mentioned in Subsection 4.4.1 that the validity of Theorem 13 for a zero-radius cycle (11)

Z_{σ_c}^s=

⎛
⎜
⎝

z	−zz
1	−z

⎞
⎟
⎠

⎛
⎜
⎝

z	z
1	1

⎞
⎟
⎠

⎛
⎜
⎝

1	−z
1	−z

⎞
⎟
⎠

with the centre z=x+i y is a straightforward calculation—see also Exercise 5. This implies the result for a generic cycle with the help of

Möbius invariance of the product (3) (and thus orthogonality) and
the above relation 4 between the orthogonality and the incidence.

The idea of such a proof is borrowed from [65] and details can be found therein.

The above demonstration suggests a generic technique for extrapolation of results from zero-radius cycles to the entire cycle space. We will formulate it with the help of a map Q from the cycle space to conics in the point space from Definition 10.

Proposition 8 Let T: ℙ³ → ℙ³ be an orthogonality-preserving map of the cycle space, i.e. ⟨ C_{σ_c}^s,S _{σ_c}^s ⟩=0 ⇔ ⟨ TC_{σ_c}^s,TS _{σ_c}^s ⟩=0. Then, for σ≠ 0, there is a map T_σ: ℝ_σ →ℝ_σ, such that Q intertwines T and T_σ:

Q T_σ= T Q. (6)

Proof. If T preserves orthogonality (i.e. the cycle product (3) and, consequently, the determinant, see (9)) then the image TZ_{σ_c}^s(u,v) of a zero-radius cycle Z_{σ_c}^s(u,v) is again a zero-radius cycle Z_{σ_c}^s(u₁,v₁). Thus we can define T_σ by the identity T_σ: (u,v)↦ (u₁,v₁).

To prove the intertwining property (6) we need to show that, if a cycle C_{σ_c}^s passes through (u,v), then the image TC_{σ_c}^s passes through T_σ(u,v). However, for σ≠ 0, this is a consequence of the T-invariance of orthogonality and the expression of the point-to-cycle incidence through orthogonality from Exercise 4.

Exercise 9 Let T_i: ℙ³ → ℙ³, i=1,2 be two orthogonality-preserving maps of the cycle space. Show that, if they coincide on the subspace of σ_c-zero-radius cycles, σ_c≠ 0, then they are identical in the whole ℙ³.

We defined orthogonality from an inner product, which is linear in each component. Thus, orthogonality also respects linearity.

Exercise 10 Check the following relations between orthogonality and pencils:

Let a cycle C_{σ_c} be σ_c-orthogonal to two different cycles S _{σ_c} and G _{σ_c}. Then C_{σ_c} is σ_c-orthogonal to every cycle in the pencil spanned by S _{σ_c} and G _{σ_c}.
Check that all cycles σ_c-orthogonal with σ_c=± 1 to two different cycles S _{σ_c} and G _{σ_c} belong to a single pencil. Describe such a family for σ_c=0. Hint: For the case σ_c=0, the family is spanned by an additional cycle, which was mentioned in Exercise 2. ⋄
If two circles are non-intersecting, then the orthogonal pencil passes through two points, which are the only two e-zero-radius cycles in the pencil, see the first row of Fig. 6.2. And vice versa: a pencil orthogonal to two intersecting circles consists of disjoint circles . Tangent circles have the orthogonal pencils of circles which are all tangent at the same point, cf. Corollary 27.
The above relations do hold for parabolas and hyperbolas only for tangent pencils (the middle column of Fig. 6.2). What are correct statements in other cases?

Exercise 2 describes two orthogonal pencils such that each cycle in one pencil is orthogonal to every cycle in the second. In terms of indefinite linear algebra, see [113]*§ 2.2, we are speaking about the orthogonal complement of a two-dimensional subspace in a four-dimensional space and it turns out to be two-dimensional as well. For circles, this construction is well known, see [71]*§ 5.7 [37]*§ 10.10. An illustration for three cases is provided by Fig. 6.2. The reader may wish to experiment more with orthogonal complements to various parabolic and hyperbolic pencils—see Fig. 5.1 and Exercise 9.

Such orthogonal pencils naturally appear in many circumstances and we already met them on several occasions. We know from Exercise 3 and 4 that K-orbits and transverse lines make coaxial pencils which turn to be in a relation:

Exercise 11 Check that:

Any K-orbit (6) in ℝ_σ is σ-orthogonal to any transverse line (8). Figure 1.2 provides an illustration. Hint: There are several ways to check this. A direct calculation based on the explicit expressions for cycles is not difficult. Alternatively, we can observe that the pencil of transverse lines is generated by K-action from the vertical axis and orthogonality is Möbius-invariant. ⋄
Any orbit (13) is σ-orthogonal to any transverse line (1) for the same subgroup K, N′ or , that is for the same value τ. Figure 3.2 provides an illustration.

The following general statement about pencils and orthogonality is an abstract generalisation of the well-known result on a triangle’s orthocenter.

Exercise 11(a) Let C_a, C_b, C_c be three pair-wise distinct cycles with at most two of them being orthogonal to each other. We define S _a as the cycle

from the pencil spanned by C_b and C_c;
orthogonal to C_a.

Cycles S _b and S _c are similarly defined. Then, S _a belongs to the pencil spanned by S _b and S _c. In particular, altitudes of a triangle are concurrent , that is meet at the same point, which is called the orthocenter of a triangle.
Hint: Note that S _a = ⟨ C_c,C_a ⟩ C_b − ⟨ C_b,C_a ⟩ C_c. ⋄

We may describe a finer structure of the cycle space through Möbius-invariant subclasses of cycles. Three such families—zero-radius, positive and negative cycles—were already considered in Sections 5.3 and 5.4. They were defined using the properties of the cycle product with itself. Another important class of cycles is given by the value of its cycle product with the real line.

Definition 12 A cycle C_{σ_c}^s is called selfadjoint if it is σ_c-orthogonal with σ_c≠ 0 to the real line, i.e. it is defined by the condition ⟨ C_{σ_c}^s,R_{σ_c}^s ⟩=0, where R_{σ_c}^s=(0,0,1,0) corresponds to the horizontal axis v=0.

The following algebraic properties of selfadjoint cycles easily follow from the definition.

Exercise 13 Show that:

Selfadjoint cycles make a Möbius-invariant family.
A selfadjoint cycle C_{σ_c}^s is defined explicitly by n=0 in (1) for both values σ_c=± 1.
Any of the following conditions are necessary and sufficient for a cycle to be selfadjoint:
- All three centres of the cycle coincide.
- At least two centres of the cycle belong to the real line.

From these analytic conditions, we can derive a geometric characterisation of selfadjoint cycles.

Exercise 14 Show that selfadjoint cycles have the following implementations in the point space ℝ_σ:

(e,h): Circles or hyperbolas with their geometric centres on the real line.
(p): Vertical lines, consisting of points (u,v) such that | u−u₀ |=r for some u₀∈ℝ, r∈ℝ₊. The cycles are also given by the ||x−y||=r² in the parabolic metric defined below in (3).

Notably, selfadjoint cycles in the parabolic point space were labelled as “parabolic circles” by Yaglom—see [339]*§ 7. On the other hand, Yaglom used the term “parabolic cycle” for our p-cycle with non-zero k and n.

Exercise 15 Show that:

Any cycle C_{σ_c}^s=(k,l,n,m) belongs to a pencil spanned by a selfadjoint cycle HC_{σ_c}^s and the real line
C_{σ_c}^s=HC_{σ_c}^s+nR_{σ_c}^s, where HC_{σ_c}^s=(k,l,0,m). (7)

This identity is a definition of linear orthogonal projection H from the cycle space to its subspace of selfadjoint cycles. When does HC_{σ_c}^s is a real cycle?
The decomposition of a cycle into the linear combination of a selfadjoint cycle and the real line is Möbius-invariant:
g·C_{σ_c}^s=g· HC_{σ_c}^s+nR_{σ_c}^s.

Hint: The first two items are small pieces of linear algebra in an indefinite product space, see [113]*§ 2.2. ⋄
Cycles C_{σ_c}^s and HC_{σ_c}^s have the same real roots.

We are now equipped to consider the geometrical meaning of all nine types of cycle orthogonality.

6.3 Ghost Cycles and Orthogonality

For the case of σ_cσ=1, i.e. when geometries of the cycle and point spaces are both either elliptic or hyperbolic, σ_c-orthogonality can be expressed locally through tangents to cycles at the intersection points—see Exercise 3(A). A semi-local condition also exists: the tangent to one cycle at the intersection point passes the centre of the second cycle—see Exercise 3(C). We may note that, in the pure parabolic case σ=σ_c=0, the geometric orthogonality condition from Exercise 3(p) can be restated with help from Exercise 1 as follows:

Corollary 16 Two p-cycles C_p and S _p are p-orthogonal if the tangent to C_p at its intersection point with the projection HS _p (7) of S _p to selfadjoint cycles passes the p-centre of S _p.

Hint: In order to reformulate Exercise 3(p) to the present form, it is enough to use Exercises 14(p) and 3. ⋄

The three cases with matching geometries in point and cycle spaces are now quite well unified. Would it be possible to extend such a geometric interpretation of orthogonality to the remaining six (=9−3) cases?

Elliptic realisations (in the point space) of Definition 1, i.e. σ=−1, were shown in Fig. 1.7 and form the first row in Fig. 6.3. The left picture in this row corresponds to the elliptic cycle space, e.g. σ_c=−1. The orthogonality between the red circle and any circle from the blue or green families is given in the usual Euclidean sense described in Exercise 3(e,h). In other words, we can decide on the orthogonality of circles by observing the angles between their tangents at the arbitrary small neighbourhood of the intersection point. Therefore, all circles from either the green or blue families, which are orthogonal to the red circle, have common tangents at points a and b respectively.

The central (parabolic in the cycle space) and the right (hyperbolic) pictures show the the non-local nature of orthogonality if σ≠σ_c. The blue family has the intersection point b with the red circle, and tangents to blue circles at b are different. However, we may observe that all of them pass the second point d. This property will be used in Section 6.5 to define the inversion in a cycle. A further investigation of Fig. 6.3 reveals that circles from the green family have a common tangent at point a, however this point does not belong to the red circle. Moreover, in line with the geometric interpretation from Exercise 3(C), the common tangent to the green family at a passes the p-centre (on the central parabolic drawing) or h-centre (on the right hyperbolic drawing).

There are analogous pictures in parabolic and hyperbolic point spaces as well and they are presented in the second and third rows of Fig. 6.3. The behaviour of green and blue families of cycles at point a, b and d is similar up to the obvious modification: the EPH case of the point space coincides with EPH case of the cycle spaces in the central drawing of the second row and the right drawing of the third row.

Therefore, we will clarify the nature of orthogonality if the locus of such points a with tangents passing the other cycle’s σ_c-centre are described. We are going to demonstrate that this locus is a cycle, which we shall call a “ghost”. The ghost cycle is shown by the dashed lines in Fig. 6.3. To give an analytic description, we need the Heaviside function χ(σ):

More specifically, we note the relations χ(σ)=σ if σ=±1 and χ(σ)=1 if σ=0. Thus, the Heaviside function will be used to avoid the degeneracy of the parabolic case.

Definition 17 For a cycle C_{σ_c} in the σ-implementation, we define the associated (σ_c-)ghost cycle G _{σ_c} by the following two conditions:

The χ(σ)-centre of G _{σ_c} coincides with the σ_c-centre of C_{σ_c}.
The determinant of G _σ¹ is equal to the determinant of C_σ^σ_c.

Exercise 18 Verify the following properties of a ghost cycle:

G _σ coincides with C_σ if σ σ_c=1.
G _σ has common roots (real or imaginary) with C_σ.
For a cycle C_{σ_c}, its p-ghost cycle G _{σ_c} and the non-selfadjoint part HC_{σ_c} (7) coincide.
All straight lines σ_c-orthogonal to a cycle pass its σ_c-centre.

The significance of the ghost cycle is that the σ_c-orthogonality between two cycles in ℝ_σ is reduced to σ-orthogonality to the ghost cycle.

Proposition 19 Let cycles C_{σ_c} and S _{σ_c} be σ_c-orthogonal in ℝ_σ and let G _{σ_c} be the ghost cycle of C_{σ_c}. Then:

S _{σ_c} and G _{σ_c} are σ-orthogonal in ℝ_σ for seven cases except two cases σ=0 and σ_c=± 1.
In the σ-implementation, the tangent line to S _{σ_c} at its intersection point with
1. the ghost cycle G _σ, if σ=± 1, or
2. the non-selfadjoint part HC_{σ_c} (7) of the cycle C_{σ_c}, if σ=0,
passes the σ_c-centre of C_{σ_c}, which coincides with the σ-centre of G _σ.

Proof. The statement 1 can be shown by algebraic manipulation, possibly in CAS. Then, the non-parabolic case 2(a) follows from the first part 1, which reduces non-matching orthogonality to a matching one with the ghost cycle, and the geometric description of matching orthogonality from Exercise 3. Therefore, we only need to provide a new calculation for the parabolic case 2(b). Note that, in the case σ=σ_c=0, there is no disagreement between the first and second parts of the proposition since HC_{σ_c}=S _{σ_c} due to 3.

Consideration of ghost cycles does present orthogonality in geometric term, but it hides the symmetry of this relation. Indeed, it is not obvious that S _{σ_c}^s relates to the ghost of C_{σ_c}^s in the same way as C_{σ_c}^s relates to the ghost of S _{σ_c}^s.

Remark 20 Elliptic and hyperbolic ghost cycles are symmetric in the real line and the parabolic ghost cycle has its centre on it—see Fig. 6.3. This is an illustration of the boundary effect from Remarks 3.

Finally, we note that Proposition 19 expresses σ_c-orthogonality through the σ_c-centre of cycles. It illustrates the meaningfulness of various centres within our approach which may not be so obvious at the beginning.

6.4 Actions of FSCc Matrices

Definition 11 associates a 2× 2-matrix to any cycle. These matrices can be treated analogously to elements of SL₂(ℝ) in many respects. Similar to the SL₂(ℝ) action (4), we can consider a fraction-linear transformation on the point space ℝ_σ defined by a cycle and its FSCc matrix

Exercise 21 Check that w=u+ι v∈ℝ_σ is a fixed point of the map C_σ^−σ (9) if and only if the σ-implementation of C_σ^−σ passes w. If detS _σ^s≠ 0 then the second iteration of the map is the identity.

We can also extend the conjugated action (7) on the cycle space from SL₂(ℝ) to cycles. Indeed, a cycle S _{σ_c}^s in the matrix form acts on another cycle C_{σ_c}^s by the σ_c-similarity

The similarity can be considered as a transformation of the cycle space ℙ³ to itself due to the following result.

Exercise 22 Check that:

The cycle σ_c-similarity (10) with a cycle S _{σ_c}^s, where det S _{σ_c}^s≠ 0, preserves the structure of FSCc matrices and S _{σ_c}^s₁ is its fixed point. In a non-singular case, detS _{σ_c}^s≠ 0, the second iteration of similarity is the identity map.
The σ_c-similarity with a σ_c-zero-radius cycle Z_{σ_c}^s always produces this cycle.

The σ_c-similarity with a cycle (k,l,n,m) is a linear transformation of the cycle space ℝ⁴ with the matrix

⎛
⎜
⎜
⎜
⎝

km−detC_{σ_c}	−2 k l	2 σ_c k n	k²
l m	−2l²−detC_{σ_c}	2 σ_c l n	k l
n m	−2 n l	2 σ_c n²−detC_{σ_c}	k n
m²	−2 ml	2σ_c m n	km−detC_{σ_c}

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎠

⎛
⎝

−2l

2σ_c n

⎞
⎠

−det(C_{σ_c})· I_4× 4 .

Note the apparent regularity of its entries.

Remark 23 Here is another example where usage of complex (dual or double) numbers is different from Clifford algebras . In order to use commutative hypercomplex numbers, we require the complex conjugation for the cycle product (3), linear-fractional transformation (9) and cycle similarity (10). Non-commutativity of Clifford algebras allows us to avoid complex conjugation in all these formulae—see Appendix B.5. For example, the reflection in the real line (complex conjugation) is given by matrix similarity with the corresponding matrix (

e₁	0
0	−e₁

A comparison of Exercises 21 and 22 suggests that there is a connection between two actions (9) and (10) of cycles, which is similar to the relation SL₂(ℝ) actions on points and cycles from Lemma 20.

Exercise 24 Letting detS _{σ_c}^s ≠ 0, show that:

The σ_c-similarity (10) σ_c-preserves the orthogonality relation (1). More specifically, if G _{σ_c}^s and G _{σ_c}^s are matrix similarity (10) of cycles C_{σ_c}^s and G _{σ_c}^s, respectively, with the cycle S _{σ_c}^s₁, then
⟨ G _{σ_c}^s,G _{σ_c}^s ⟩= ⟨ C_{σ_c}^s,G _{σ_c}^s ⟩ (detS _{σ_c}^s)².

Hint: Note that S _{σ_c}^sS _{σ_c}^s= −det(S _{σ_c}^s) I, where I is the identity matrix. This is a particular case of the Vahlen condition, see [100]*Prop. 2. Thus, we have
G _{σ_c}^s

G _{σ_c}^s

= −S _{σ_c}^s₁C_{σ_c}^s

G _{σ_c}^s

S _{σ_c}^s₁

· det S _{σ_c}^s.

The final step uses the invariance of the trace under the matrix similarity. A CAS calculation is also provided. ⋄
The image T_σ^s=C_σ^s₂ Z_σ^s₁C_σ^s₂ of a σ-zero-radius cycle T_σ^s₁ under the similarity (10) is a σ-zero-radius cycle T_σ^s₁. The (s₁s₂)-centre of T_{σ_c}^s is the linear-fractional transformation (9) of the (s₂/s₁)-centre of Z_{σ_c}^s.
Both formulae (9) and (10) define the same transformation of the point space ℝ_σ, with σ=σ_c≠ 0. Consequently, the linear-fractional transformation (9) maps cycles to cycles in these cases. Hint: This part follows from the first two items and Proposition 8. ⋄
There is a cycle C_σ^s such that neither map of the parabolic point space ℝ_p represents similarity with C_σ^s. Hint: Consider S _σ^s=(1,0,1/2,−1) and a cycle C_σ^s passing point (u,v). Then the similarity of C_σ^s with S _σ^s passes the point T(u,v)=(1+v/u,v+v²/u²) if and only if either
- C_σ^s is a straight line, or
- (u,v) belongs to S _σ^s and is fixed by the above map T.
That is, the map T of the point space ℝ_p serves flat cycles and S _σ^s but no others. Thus, there is no map of the point space which is compatible with the cycle similarity for an arbitrary cycle. ⋄

Several demonstrations of inversion are provided in Fig. 6.4. The initial setup is shown in Fig. 6.4(a)—the red unit circle and the grid of horizontal (green) and vertical (blue) straight lines. It is very convenient in this case that the grid is formed by two orthogonal pencils of cycles , which can be considered to be of any EPH type. Figure 6.4(b) shows e-inversion of the grid in the unit circle, which is, of course, the locus of fixed points. Straight lines of the rectangular grid are transformed to circles, but orthogonality between them is preserved—see Exercise 1.

Similarly, Fig. 6.4(c) presents the result of p-inversion in the degenerated parabolic cycle u²−1=0. This time the grid is mapped to two orthogonal pencils of parabolas and vertical lines. Incidentally, due to the known optical illusion, we perceive these vertical straight lines as being bent.

Finally, Fig. 6.4(d) demonstrates h-inversion in the unit hyperbola u²−v²−1=0. We again obtained two pencils of orthogonal hyperbolas. The bold blue cycles—the dot at the origin in (b), vertical line in (c) and two lines (the light cone ) in (d)—will be explained in Section 8.1. Further details are provided by the following exercise.

Exercise 25 Check that the rectangular grid in Fig. 6.4(a) is produced by horizontal and vertical lines given by quadruples (0,0,1,m) and (0,1,0,m), respectively.

The similarity with the cycle (1,0,0,−1) sends a cycle (k,l,n,m) to (m,l,n,k). In particular, the image of the grid are cycles (m,0,1,0) and (m,1,0,0).

We conclude this section by an observation that cycle similarity is similar to a mirror reflection, which preserves the directions of vectors parallel to the mirror and reverses vectors which are orthogonal.

Exercise 26 Let detS _{σ_c}^s≠ 0. Then, for similarity (10) with S _{σ_c}^s:

Verify the identities
−S _{σ_c}^s₁

S _{σ_c}^s

S _{σ_c}^s₁

=

det

σ_c

(S _{σ_c}^s₁)· S _{σ_c}^s₁ and

−S _{σ_c}

C_{σ_c}

S _{σ_c}

=
−

det

σ_c

(S _{σ_c})·C_{σ_c},

where C_{σ_c}^s is a cycle σ_c-orthogonal to S _{σ_c}. Note the difference in the signs in the right-hand sides of both identities.
Describe all cycles which are fixed (as points in the projective space ℙ³) by the similarity with the given cycle S _{σ_c}^s. Hint: Use a decomposition of a generic cycle into a sum S _{σ_c}^s and a cycle orthogonal to S _{σ_c}^s similar to (7). ⋄

As we will see in the next section, these orthogonal reflections in the cycle space correspond to “bent” reflections in the point space.

6.5 Inversions and Reflections in Cycles

The maps in point and cycle spaces considered in the previous section were introduced from the action of FSCc matrices of cycles. They can also be approached from the more geometrical viewpoint. There are at least two natural ways to define an inversion in a cycle :

Definition 27 For a given cycle C_σ^s, we define two maps of the point space ℝ_σ associated to it:

A σ_c-inversion in a σ-cycle C_σ^s sends a point b∈ℝ_σ to the second point d of the intersection of all σ-cycles σ_c-orthogonal to C_σ^s and passing through b—see Fig. 6.3.
A σ_c-reflection in a σ-cycle C_σ^s is given by M⁻¹RM, where M is a σ_c-similarity (10) which sends the σ-cycle C_σ^s into the horizontal axis and R is the mirror reflection of ℝ_σ in that axis.

We are going to see that inversions are given by (9) and reflections are expressed through (10), thus they are essentially the same for EH cases in light of Exercise 1. However, some facts are easier to establish using the inversion and others in terms of reflection. Thus, it is advantageous to keep both notions. Since we have three different EPH orthogonalities between cycles for every type of point space, there are also three different inversions in each of them.

Exercise 28 Prove the following properties of inversion:

Let a cycle S _{σ_c}^s be σ_c-orthogonal to a cycle C_{σ_c}^s=(k,l,n,m). Then, for any point u₁+ι v₁∈ ℝ_σ (ι²=σ) belonging to σ-implementation of S _{σ_c}^s, this implementation also passes through the image of u₁+ι v₁ under the Möbius transform (9) defined by the matrix C_σ^σ_c:
u₂+ι v₂ =C_σ^σ_c(u₁−ι v₁ )=
⎛
⎜
⎝
l+ισ_c n −m

k −l+ισ_c n

⎞
⎟
⎠

(u₁−ι v₁). (11)

Thus, the point u₂+ι v₂ is the inversion of u₁+ι v₁ in C_{σ_c}^s.
Conversely, if a cycle S _{σ_c}^s passes two different points u₁+ι v₁ and u₂+ι v₂ related through (11), then S _{σ_c}^s is σ_c-orthogonal to C_{σ_c}^s.
If a cycle S _{σ_c}^s is σ_c-orthogonal to a cycle C_{σ_c}^s, then the σ_c-inversion in C_{σ_c}^s sends S _{σ_c}^s to itself.
σ_c-inversion in the σ-implementation of a cycle C_{σ_c}^s coincides with σ-inversion in its σ_c-ghost cycle G _{σ_c}^s.

Note the interplay between parameters σ and σ_c in the above statement 1. Although we are speaking about σ_c-orthogonality, we take the Möbius transformation (11) with the imaginary unit ι such that ι²=σ (as the signature of the point space). On the other hand, the value σ_c is used there as the s-parameter for the cycle C_σ^σ_c.

Proposition 29 The reflection 2 of a zero-radius cycle Z_{σ_c}^s in a cycle C_{σ_c}^s is given by the similarity C_{σ_c}^sZ_{σ_c}^sC_{σ_c}^s.

Proof. Let a cycle S _{σ_c}^s have the property S _{σ_c}^s C_{σ_c}^s S _{σ_c}^s = R_{σ_c}^s, where R_{σ_c}^s is the cycle representing the real line. Then, S _{σ_c}^s R_{σ_c}^s S _{σ_c}^s = C_{σ_c}^s, since S _{σ_c}^sS _{σ_c}^s= S _{σ_c}^sS _{σ_c}^s = −det S _{σ_c}^sI. The mirror reflection in the real line is given by the similarity with R_{σ_c}^s, therefore the transformation described in 2 is a similarity with the cycle S _{σ_c}^s R_{σ_c}^s S _{σ_c}^s = C_{σ_c}^s and, thus, coincides with (11).

Corollary 30 The σ_c-inversion with a cycle C_{σ_c}^s in the point space ℝ_σ coincides with σ_c-reflection in C_{σ_c}^s.

The auxiliary cycle S _{σ_c}^s from the above proof of Prop. 29 is of separate interest and can be characterised in the elliptic and hyperbolic cases as follows.

Exercise 31 Let C_{σ_c}^s=(k, l, n,m) be a cycle such that σ_c detC_{σ_c}^s>0 for σ_c≠ 0. Let us define the cycle S _{σ_c}^s by the quadruple (k, l, n±√σ_c detC_{σ_c}^s,m). Then:

S _{σ_c}^s C_{σ_c}^s S _{σ_c}^s = ℝ and S _{σ_c}^s ℝ S _{σ_c}^s = C_{σ_c}^s.
S _{σ_c}^s and C_{σ_c}^s have common roots.
In the σ_c-implementation, the cycle C_{σ_c}^s passes the centre of S _{σ_c}^s.

Hint: One can directly observe 2 for real roots, since they are fixed points of the inversion. Also, the transformation of C_{σ_c}^s to a flat cycle implies that C_{σ_c}^s passes the centre of inversion, hence 3. There is also a CAS calculation for this. ⋄

Inversions are helpful for transforming pencils of cycles to the simplest possible form.

Exercise 32 Check the following:

Let the σ-implementation of a cycle C_σ^s pass the σ-centre of a cycle S _σ^s. Then, the σ-reflection of C_σ^s in S _σ^s is a straight line.
Let two cycles C_σ^s and S _σ^s intersect in two points P, P′∈ℝ_σ such that P−P′ is not a divisor of zero in the respective number system. Then, there is an inversion which maps the pencil of cycles orthogonal to C_σ^s and S _σ^s (see Exercise 2) into a pencil of concentric cycles. Hint: Make an inversion into a cycle with σ-centre P, then C_σ^s and S _σ^s will be transformed into straight lines due to the previous item. These straight lines will intersect in a finite point P″ which is the image of P′ under the inversion. The pencil orthogonal to C_σ^s and S _σ^s will be transformed to a pencil orthogonal to these two straight lines. A CAS calculations shows that all cycles from the pencil have σ-centre at P″. ⋄

A classical source of the above result in inversive geometry [71]*Thm. 5.71 tells that an inversion can convert any pair of non-intersecting circles to concentric ones. This is due to the fact that an orthogonal pencil to the pencil generated by two non-intersecting circles always passes two special points—see Exercise 3 for further development.

Finally, we compare our consideration for the parabolic point space with Yaglom’s book. The Möbius transformation (9) and the respective inversion illustrated by Fig. 6.4(c) essentially coincide with the inversion of the first kind from [339]*§ 10. Yaglom also introduces the inversion of the second kind, see [339]*§ 10. For a parabola v=k(u−l)²+m, he defined the map of the parabolic point space to be

i.e. the parabola bisects the vertical line joining a point and its image. There are also other geometric characterisations of this map in [339], which make it very similar to the Euclidean inversion in a circle. Here is the resulting expression of this transformation through the usual inversion in parabolas:

Exercise 33 The inversion of the second kind (12) is a composition of three Möbius transformations (9) defined by cycles (1,l,2m,l²+m/k), (1,l,0,l²+m/k) and the real line in the parabolic point space ℝ_p.

Möbius transformations (9) and similarity (12) with FSCc matrices map cycles to cycles just like matrices from SL₂(ℝ) do. It is natural to ask for a general type of matrices sharing this property. For this, see works [65, 100, 289] which deal with more general elliptic and hyperbolic (but not parabolic) cases. It is beyond the scope of our consideration since it derails from the geometry of SL₂(ℝ). We only mention the rôle of the Vahlen condition, C_{σ_c}^sC_{σ_c}^s= −det(C_{σ_c}^s) I, used in Exercise 1.

6.6 Higher-order Joint Invariants: Focal Orthogonality

Considering Möbius action (1), there is no need to be restricted to joint invariants of two cycles and a bilinear form. Indeed, for any polynomial p(x₁,x₂,…,x_n) of several non-commuting variables, one may define an invariant joint disposition of n cycles ^jC_{σ_c}^s by the condition

where the polynomial of FSCc matrices is defined through standard matrix algebra. To create a Möbius invariant which is not affected by the projectivity in the cycle space we can either

Furthermore, for any two polynomials p(x₁,x₂,…,x_n) and q(x₁,x₂,…,x_n) such that for each variable x_i the orders of homogeneity of p and q are equal, the value

is a Möbius invariant. The simplest and yet most important realisation of this concept is the cycles cross ratio in § 12.3

Let us construct some lower-order realisations of (13). In order to be essentially different from the previously considered orthogonality, such invariants may either contain non-linear powers of the same cycle, or accommodate more than two cycles. In this respect, consideration of higher-order invariants is similar to a transition from Riemannian to Finsler geometry [62, 107]. The latter is based on the replacement of the quadratic line element g^ij dx_i dx_j in the tangent space by a more complicated function.

A further observation is that we can simultaneously study several invariants of various orders and link one to another by some operations. There are some standard procedures changing orders of invariants working in both directions:

Consider both operations as an example. We already know that a similarity of a cycle with another cycle produces a new cycle. The cycle product of the latter with a third cycle creates a joint invariant of these three cycles

which is built from the second-order invariant ⟨ ·,· ⟩. Now we can reduce the order of this invariant by fixing ³C_{σ_c}^s to be the real line, since it is SL₂(ℝ)-invariant. This invariant deserves special consideration. Its geometrical meaning is connected to the matrix similarity of cycles (10) (inversion in cycles) and orthogonality.

Definition 34 A cycle S _{σ_c}^s is σ_c-focal orthogonal (or f_{σ_c}-orthogonal) to a cycle C_{σ_c}^s if the σ_c-reflection of C_{σ_c}^s in S _{σ_c}^s is σ_c-orthogonal (in the sense of Definition 1) to the real line. We denote it by S _{σ_c}^s ⊣ C_{σ_c}^s.

Remark 35 This definition is explicitly based on the invariance of the real line and is an illustration to the boundary value effect from Remark 3.

Exercise 36 f-orthogonality is equivalent to either of the following

The cycle S _{σ_c}^s C_{σ_c}^sS _{σ_c}^s is a selfadjoint cycle , see Definition 12.
Analytical condition:
⟨ S _{σ_c}^s

C_{σ_c}^s

S _{σ_c}^s,R_{σ_c}^s ⟩= tr(S _{σ_c}^s

C_{σ_c}^s

S _{σ_c}^sR_{σ_c}^s)=0. (15)

Remark 37 It is easy to observe the following:

f-orthogonality is not symmetric: C_{σ_c}^s ⊣ S _{σ_c}^s does not imply S _{σ_c}^s ⊣ C_{σ_c}^s.
Since the horizontal axis R_{σ_c}^s and orthogonality (1) are SL₂(ℝ)-invariant objects, f-orthogonality is also SL₂(ℝ)-invariant.

However, an invariance of f-orthogonality under inversion of cycles required some study since, in general, the real line is not invariant under such transformations.

Exercise 38 The image G _{σ_c}^s₁ R_{σ_c}^s G _{σ_c}^s₁ of the real line under inversion in G _{σ_c}^s₁=(k,l,n,m) with s≠ 0 is the cycle

(2 s s₁ σ_c k n, 2 s s₁ σ_c l n, −det(C_{σ_c}^s₁), 2 s s₁ σ_c m n).

It is the real line again if det(C_{σ_c}^s₁)≠0 and either

s₁ n=0, in which case it is a composition of SL₂(ℝ)-action by (
l −m

k −l

) and the reflection in the real line, or
σ_c=0, i.e. the parabolic case of the cycle space.

If either of two conditions is satisfied then f-orthogonality S _{σ_c}^s⊣ C_{σ_c}^s is preserved by the σ_c-similarity with G _{σ_c}^s₁.

The following explicit expressions of f-orthogonality reveal further connections with cycles’ invariants.

Exercise 39 f-orthogonality of S _{σ_c}^s to C_{σ_c}^s₁ is given by either of the equivalent identities

s n (l′²+σ_c s₁²n′²− m′ k′) + s₁n′(m k′ −2l l′+ k m′ )	=	0 or
n det(S _{σ_c}¹) + n′⟨ C_{σ_c}¹,S _{σ_c}¹ ⟩	=	0, if s=s₁=1.

The f-orthogonality may again be related to the usual orthogonality through an appropriately chosen f-ghost cycle, cf. Proposition 19:

Proposition 40 Let C_{σ_c}^s be a cycle. Then, its f-ghost cycle G _{σ_c}^σ_c = C_{σ_c}^χ(σ) ℝ_{σ_c}^σ_c C_{σ_c}^χ(σ) is the reflection of the real line in C_{σ_c}^χ(σ), where χ(σ) is the Heaviside function (8). Then:

Cycles C_{σ_c}¹ and G _{σ_c}^σ_c have the same roots.
The χ(σ)-centre of G _{σ_c}^σ_c coincides with the (−σ_c)-focus of C_{σ_c}^s, consequently all straight lines σ_c-f-orthogonal to C_{σ_c}^s pass its (−σ_c)-focus.
f-inversion in C_{σ_c}^s defined from f-orthogonality (see Definition 1) coincides with the usual inversion in G _{σ_c}^σ_c.

Note the above intriguing interplay between the cycle’s centres and foci. It also explains our choice of name for focal orthogonality, cf. Definition 1. f-Orthogonality and the respective f-ghost cycles are presented in Fig. 6.5, which uses the same outline and legend as Fig. 6.3.

The definition of f-orthogonality may look rather extravagant at first glance. However, it will find new support when we again consider lengths and distances in the next chapter. It will also be useful for infinitesimal cycles, cf. Section 7.5.

Of course, it is possible and meaningful to define other interesting higher-order joint invariants of two or even more cycles.

Lecture 6 Joint Invariants of Cycles: Orthogonality

6.1 Orthogonality of Cycles

6.2 Orthogonality Miscellanea

6.3 Ghost Cycles and Orthogonality

6.4 Actions of FSCc Matrices

6.5 Inversions and Reflections in Cycles

6.6 Higher-order Joint Invariants: Focal Orthogonality