Metric Invariants in Upper Half-Planes

So far, we have discussed only invariants like orthogonality, which are related to angles. However, geometry, in the plain meaning of the word, deals with measurements of distances and lengths. We will derive metrical quantities from cycles in a way which shall be Möbius-invariant.

7.1 Distances

Cycles are covariant objects performing as “circles” in our three EPH geometries. Now we play the traditional mathematical game: turn some properties of classical objects into definitions of new ones.

Definition 1 The σ_c-radius r_{σ_c} of a cycle C_{σ_c}^s, if squared, is equal to the minus σ_c-determinant of the cycle’s k-normalised (see Definition 5) matrix, i.e.

r_{σ_c}²= −

⟨ C_{σ_c}^s,C_{σ_c}^s ⟩

2k²

= −

detC_{σ_c}^s

k²

l²− σ_c s²n²−km

k²

. (1)

As usual, the σ_c-diameter of a cycle is two times its radius.

The expression (1) for radius through the invariant cycle product resembles the definition of the norm of a vector in an inner product space [164]*§ 5.1.

Exercise 2 Check the following geometrical content of the formula (1):

(e,h): The value of (1) is the usual radius of a circle or hyperbola given by the equation k(u²−σ v²)−2lu−2nv+m=0.
(p): The diameter of a parabola is the (Euclidean) distance between its (real) roots, i.e. solutions of ku²−2lu+m=0, or roots of its “adjoint” parabola −ku²+2lu+m−2l²/k=0 (see Fig. 1.9(a)).

Exercise 3 Check the following relations:

The σ_c-radius of a cycle C_{σ_c}^s is equal to 1/k, where k is the (2,1)-entry of the det-normalised FSCc matrix (see Definition 5) of the cycle.
Let u_{σ_c} be the second coordinate of a cycle’s σ_c-focus and f be its focal length . Then, the square of the cycle’s σ_c-radius is
r_{σ_c}²=−4fu_{σ_c}.
Cycles (11) have zero σ_c-radius, thus Definitions 13 and 1 agree.

An intuitive notion of a distance in both mathematics and everyday life is usually of a variational nature. We naturally perceive a straight line as the route of the shortest distance between two points. Then, we can define the distance along a curved path through an approximation. This variational nature is also echoed in the following definition:

Definition 4 The (σ,σ_c)-distance d_{σ,σ_c}(P,P′) between two points P and P′ is the extremum of σ_c-diameters for all σ-implementations of cycles passing P and P′.

Lemma 5 The distance between two points P=u+ι v and P′=u′+ι v′ in the elliptic or hyperbolic spaces is

d_{σ,σ_c}²(P, P′) =

σ_c ((u−u′)²−σ(v− v′)²) +4(1−σσ_c) v v′

(u− u′)² σ_c−(v−v′)²

((u−u′)² −σ(v− v′)²), (2)

and, in parabolic space, it is (see Fig. 1.9(b) and also [339]*(5), p. 38)

d_{p,σ_c}²(y, y′) = (u−u′)². (3)

Proof. Let C_s^σ(l) be the family of cycles passing through both points (u, v) and (u′, v′) (under the assumption v≠ v′) and parametrised by its coefficient l, which is the first coordinate of the cycle’s centre. By a symbolic calculation in CAS, we found that the only critical point of det(C_s^σ(l)) is

Note that, for the case σσ_c=1, i.e. when both point and cycle spaces are simultaneously either elliptic or hyperbolic, this expression reduces to the expected midpoint l₀=1/2(u+u′). Since, in the elliptic or hyperbolic case, the parameter l can take any real value, the extremum of det(C_s^σ(l)) is reached in l₀ and is equal to (2), again calculated by CAS. A separate calculation for the case v= v′ gives the same answer.

In the parabolic case, possible values of l are either in the ranges (−∞, 1/2(u+u′)) or (1/2(u+u′),∞), or only l=1/2(u+u′) since, for that value, a parabola should flip between the upward and downward directions of its branches. In any of these cases, the extreme value corresponds to the boundary point l=1/2(u+u′) and is equal to (3).

These are familiar expressions for distances in the elliptic and hyperbolic spaces. However, for other cases (such that σσ_c=−1 or 0) quite different results are obtained. For example, d_{σ,σ_c}²(P, P′) does not tend to 0 if P→ P′ in the usual sense.

Exercise 6 Show that:

In the three cases σ=σ_c=−1, 0 or 1, which were typically studied in the literature, the above distances are conveniently defined through the arithmetic of corresponding numbers:
d_σ,σ²(u+ι v)=(u+vι)(u−vι)=ww. (5)
Unless σ=σ_c, the parabolic distance d_{p,σ_c} (3) is not obtained from (2) by the substitution σ =0.
If cycles C_{σ_c}^s and S _{σ_c}^s are k-normalised , then
⟨ C_{σ_c}^s−S _{σ_c}^s,C_{σ_c}^s−S _{σ_c}^s ⟩ =2det(C_{σ_c}^s−S _{σ_c}^s) =−2d_{σ_c,σ_c}²(P,P′),

where P and P′ are e- or h-centres of C_{σ_c}^s and S _{σ_c}^s. Therefore, we can rewrite the relation (8) for the cycle product
⟨ C_{σ_c}^s,S _{σ_c}^s ⟩ = d_{σ_c,σ_c}²(P,P′)−r_{σ_c}²−r′_{σ_c}²,

using r_{σ_c} and r′_{σ_c}—the σ_c-radii of the respective cycles. In particular, cf. (12),
⟨ Z_{σ_c}^s,T_{σ_c}^s ⟩ = d_{σ_c,σ_c}²(P,P′),

for k-normalised σ_c-zero-radius cycles Z_{σ_c}^s and T_{σ_c}^s with centres P and P′. From Exercise 21 we can also derive that d_{σ_c,σ_c}²(P,P′) is equal to the power of the point P (P′) with respect to the cycle T_{σ_c}^s (Z_{σ_c}^s).

Using the notation d_{σ ,σ}²(P)=d_{σ ,σ}²(0,P), we can now rewrite the identities (3)–(4) as:

The distance allows us to expand the result for all EPH cases, which is well known in the cases of circles [71]*§ 2.1 and parabolas [339]*§ 10.

Exercise 7 Show that the power of a point W with respect to a cycle (see Definition 18) is the product d_σ(W,P)· d_σ(W,P′) of (σ,σ)-distances (5), where P and P′ are any two points of the cycle which are collinear with W. Hint: Take P=(u,v) and P′=(u′,v′). Then, any collinear W is (tu+(1−t)u′,tv+(1−t)v′) for some t∈ℝ. Furthermore, a simple calculation shows that d_σ(y,z)· d_σ(z,y′)= t(t−1)d_σ²(y,y′). The last expression is equal to the power of W with respect to the cycle—this step can be done by CAS. ⋄

Exercise 8 Let two cycles have e-centres P and P′ with σ_c-radii r_{σ_c} and r_{σ_c}′. Then, the (σ_c,σ_c)-power of one cycle with respect to another from Definition 22 is, cf. the elliptic case in (13),

d_{σ_c,σ_c}²(P,P′)−r_{σ_c}²−r_{σ_c}^′ 2

2r_{σ_c}  r_{σ_c}′

7.2 Lengths

During geometry classes, we often make measurements with a compass, which is based on the idea that a circle is a locus of points equidistant from its centre. We can expand it for all cycles with the following definition:

Definition 9 The (σ,σ_c,σ_r)-length from the σ_r-centre (from the σ_r-focus ) of a directed interval AB is the σ_c-radius of the σ-cycle with its σ_r-centre (σ_r-focus) at the point A which passes through B. These lengths are denoted by l_c(AB) and l_f(AB), respectively.

It is easy to be confused by the triple of parameters σ, σ_c and σ_r in this definition. However, we will rarely operate in such a generality, and some special relations between the different sigmas will often be assumed. We also do not attach the triple (σ,σ_c,σ_r) to l_c(AB) and l_f(AB) in formulae, since their values will be clear from the surrounding text.

Exercise 10 Check the following properties of the lengths:

The length is not a symmetric function of two points (unlike the distance).
A cycle is uniquely defined by its elliptic or hyperbolic centre and a point which it passes. However, the parabolic centre is not as useful. Consequently, lengths from the parabolic centre are not properly defined, therefore we always assume σ_r=± 1 for lengths from a centre.
A cycle is uniquely defined by any focus and a point which it passes.

Lemma 11 For two points P=u+ι v, P′=u′+ι v′∈ℝ_σ:

The σ_c-length from the σ_r-centre for σ_r=±1 between P and P′ is
l_{c_{σ_c}}²(
▸

PP′

) = (u−u′)²−σ v′²+2σ_r v v′ −σ_c v². (6)

The σ_c-length from the σ_r-focus between P and P′ is

l_{f_{σ_c}}²(

	▸
PP′

) = (σ_r−σ_c) p²−2vp, (7)

where:

σ_r

⎛
⎜
⎝

−(v′−v)±

√

σ_r(u′−u)²+(v′−v)²−σσ_r v′²

⎞
⎟
⎠

, if σ_r≠0

(8)

(u′−u)²−σ v′²

2(v′−v)

, if σ_r=0.

(9)

Proof. Identity (6) is verified with CAS. For the second part, we observe that a cycle with the σ_r-focus (u,v) passing through (u′,v′)∈ℝ_σ has the parameters:

Exercise 12 Check that:

The value of p in (8) is the focal length of either of the two cycles, which are, in the parabolic case, upward or downward parabolas (corresponding to the plus or minus signs) with focus at (u, v) and passing (u′, v′).
In the case σσ_c=1, the length from the centre (6) becomes the standard elliptic or hyperbolic distance (u−u′)²−σ (v−v′)² obtained in (2). Since these expressions appeared both as distances and lengths they are widely used.
On the other hand, in the parabolic point space, we obtain three additional lengths besides distance (3):
l_{c_{σ_c}}²(y, y′) = (u−u′)²+2 v v′−σ_c v²

parametrised by three values −1, 0 or −1 of σ_c (cf. Remark 1).
The parabolic distance (3) can be expressed in terms of the focal length p (8) as
d²(y, y′) = p²+2(v−v′)p,

an expression similar to (7).

7.3 Conformal Properties of Möbius Maps

All lengths l(AB) in ℝ_σ from Definition 9 are such that, for a fixed point A, every contour line l(AB)=c is a corresponding σ-cycle, which is a covariant object in the appropriate geometry. This is also true for distances if σ=σ_c. Thus, we can expect some covariant properties of distances and lengths.

Definition 13 We say that a distance or a length d is SL₂(ℝ)-conformal if, for fixed y, y′∈ℝ_σ, the limit

lim

t→ 0

d(g· y, g·(y+ty′))

d(y, y+ty′)

, where g∈SL₂(ℝ), (10)

exists and its value depends only on y and g and is independent of y′.

Informally rephrasing this definition, we can say that a distance or length is SL₂(ℝ)-conformal if a Möbius map scales all small intervals originating at a point by the same factor. Also, since a scaling preserves the shape of cycles, we can restate the SL₂(ℝ)-conformality once more in familiar terms: small cycles are mapped to small cycles. To complete the analogy with conformality in the complex plane we note that preservation of angles (at least orthogonality) by Möbius transformations is automatic.

Exercise 14 Show SL₂(ℝ)-conformality in the following cases:

The distance (2) is conformal if and only if the type of point and cycle spaces are the same, i.e. σσ_c=1. The parabolic distance (3) is conformal only in the parabolic point space.
The lengths from centres (6) are conformal for any combination of values of σ, σ_c and σ_r.
The lengths from foci (7) are conformal for σ_r≠ 0 and any combination of values of σ and σ_c.

The conformal property of the distance (2) and (3) from Proposition 1 is, of course, well known (see [65, 339]). However, the same property of non-symmetric lengths from Proposition 2 and 3 could hardly be expected. As a possible reason, one remarks that the smaller group SL₂(ℝ) (in comparison to all linear-fractional transforms of the whole ℝ²) admits a larger number of conformal metrics, cf. Remark 3.

The exception of the case σ_r=0 from the conformality in 3 looks disappointing at first glance, especially in the light of the parabolic Cayley transform considered later in Section 10.3. However, a detailed study of algebraic structure invariant under parabolic rotations, see Chapter 11, will remove obscurity from this case. Indeed, our Definition 13 of conformality heavily depends on the underlying linear structure in ℝ_σ—we measure a distance between points y and y+ty′ and intuitively expect that it is always small for small t. As explained in Section 11.4, the standard linear structure is incompatible with the parabolic rotations and thus should be replaced by a more relevant one. More precisely, instead of limits y′→ y along the straight lines towards y, we need to consider limits along vertical lines, as illustrated in Fig. 10.1 and Definition 23.

We will return to the parabolic case of conformality in Proposition 24. An approach to the parabolic point space and a related conformality based on infinitesimal cycles will be considered in Section 7.6.

Remark 15 The expressions of lengths (6) and (7) are generally non-symmetric and this is a price one should pay for their non-triviality. All symmetric distances lead only to nine two-dimensional Cayley–Klein geometries, see [339]*App. B [284] [116] [130] [131]. In the parabolic case, a symmetric distance of a vector (u,v) is always a function of u alone, cf. Remark 21. For such a distance, a parabolic unit circle consists of two vertical lines (see dotted vertical lines in the second rows in Figs 6.3 and 6.5), which is not aesthetically attractive. On the other hand, the parabolic “unit cycles” defined by lengths (6) and (7) are parabolas, which makes the parabolic Cayley transform (see Section 10.3) very natural.

We can also consider a distance between points in the upper half-plane which has a stronger property than SL₂(ℝ)-conformality. Namely, the metric shall be preserved by SL₂(ℝ) action or, in other words, Möbius transformations are isometries for it. We will study such a metric in Chapter 9.

7.4 Perpendicularity and Orthogonality

In a Euclidean space , the shortest distance from a point to a straight line is provided by the corresponding perpendicular. Since we have already defined various distances and lengths, we may use them for a definition of the respective notions of perpendicularity¹.

Definition 16 Let l be a length or a distance. We say that a vector AB is l-perpendicular to a vector CD if function l(AB+t CD) of a variable t has a local extremum at t=0, cf. Fig. 1.10. This is denoted by AB⋋_l CD or, simply, AB⋋ CD if the meaning of l is clear from the context.

Exercise 17 Check that:

l-perpendicularity is not a symmetric notion (i.e. AB⋋ CD does not imply CD⋋ AB), similar to f-orthogonality—see Section 6.6.
l-perpendicularity is linear in CD, i.e. AB⋋ CD implies AB⋋ rCD for any real non-zero r. However, l-perpendicularity is not generally linear in AB, i.e. AB⋋ CD does not necessarily imply rAB⋋ CD.

Lemma 18 Let AB be l_c-perpendicular (l_f-perpendicular) to a vector CD for a length from the centre (from the focus) defined by the triple (σ,σ_c,σ_r). Then, the flat cycle (straight line) AB is σ_r-(f-)orthogonal to the σ-cycle C_σ^s with σ_r-centre ((−σ_r)-focus) at A passing through B. The vector CD is tangent at B to the σ -implementation of C_σ^s.

Proof. Consider the cycle C_σ^s with its σ_r-centre at A and passing B in its σ-implementation. This cycle C_σ^s is a contour line for a function l(X)=l_c(AX) with the triple (σ,σ_c,σ_r). Therefore, the cycle separates regions where l(X)< l_c(AB) and l(X)> l_c(AB). The tangent line to C_σ^s at B (or, at least, its portion in the vicinity of B) belongs to one of these two regions, thus l(X) has a local extremum at B. Therefore, AB is l_c-perpendicular to the tangent line. The line AB is also σ_r-orthogonal to the cycle C_{σ_c}^s since it passes its σ_r-centre A, cf. Exercise 4.

The second case of l_f(AB) and f-orthogonality can be considered similarly with the obvious change of centre to focus, cf. Proposition 2.

Obviously, perpendicularity turns out to be familiar in the elliptic case, cf. Lemma 1 below. For the two other cases, the description is given as follows:

Exercise 19 Let A=(u,v) and B=(u′,v′). Then

d-perpendicular (in the sense of (2)) to AB in the elliptic or hyperbolic cases is a multiple of the vector

⎛
⎜
⎝

σ (v−v′)³−(u−u′)² (v+v′(1−2 σ σ_c))

σ_c(u−u′)³−(u−u′)(v− v′)(−2 v′ +(v+v′) σ_c σ)

⎞
⎟
⎠

which, for σσ_c=1, reduces to the expected value (v−v′, σ(u−u′)).

d-perpendicular (in the sense of (3)) to AB in the parabolic case is (0, t), t∈ℝ which coincides with the Galilean orthogonality defined in [339]*§ 3.
l_{c_{σ_c}}-perpendicular (in the sense of (6)) to AB is a multiple of (σ v′−σ_r v, u−u′).
l_{f_{σ_c}}-perpendicular (in the sense of (7)) to AB is a multiple of (σ v′+p, u−u′), where p is defined either by (8) or (9) for corresponding values of σ_r.

Hint: Contour lines for all distances and lengths are cycles. Use implicit derivation of (1) to determine the tangent vector to a cycle at a point. Then apply the formula to a cycle which passes (u′,v′) and is a contour line for a distance or length from (u,v). ⋄

It is worth having an idea about different types of perpendicularity in terms of the standard Euclidean geometry . Here are some examples.

Exercise 20 Let AB=u +ι v and CD=u′+ι v′. Then:

In the elliptic case, d-perpendicularity for σ_c=−1 means that AB and CD form a right angle, or, analytically, u u′+v v′=0.
In the parabolic case, l_{f_{σ_c}}-perpendicularity for σ_c=1 means that AB bisects the angle between CD and the vertical direction, or, analytically,
u′u−v′p=u′u−v′( √

u²+v²

−v)=0, (11)

where p is the focal length (8).
In the hyperbolic case, d-perpendicularity for σ_c=−1 means that the angles between AB and CD are bisected by lines parallel to u=± v, or, analytically, u′ u−v′ v=0. Compare with Exercise 3(A).

Remark 21 If one attempts to devise a parabolic length as a limit or an intermediate case for the elliptic l_e=u²+v² and hyperbolic l_p=u²−v² lengths, then the only possible guess is l′_p=u² (3), which is too trivial for an interesting geometry.

Similarly, the only orthogonality condition linking the elliptic u₁ u₂+v₁ v₂=0 and the hyperbolic u₁ u₂−v₁ v₂=0 cases from the above exercise seems to be u₁ u₂=0 (see [339]*§ 3 and 2), which is again too trivial. This supports Remark 2.

7.5 Infinitesimal-radius Cycles

Although parabolic zero-radius cycles defined in 13 do not satisfy our expectations for “smallness”, they are often technically suitable for the same purposes as elliptic and hyperbolic ones. However, we may want to find something which fits our intuition about “zero-sized” objects better. Here, we present an approach based on non-Archimedean (non-standard) analysis —see, for example, [73, 328] for a detailed exposition.

Apart from this inequality infinitesimals obey all other properties of real numbers. In particular, in our CAS computations, an infinitesimal will be represented by a positive real symbol and we replace some of its powers by zero if their order of infinitesimality will admit this. The existence of infinitesimals in the standard real analysis is explicitly excluded by the Archimedean axiom, therefore the theory operating with infinitesimals is known as non-standard or non-Archimedean analysis. We assume from now on that there exists an infinitesimal number є.

Definition 22 A cycle C_{σ_c}^s, such that detC_{σ_c}^s is an infinitesimal number, is called an infinitesimal radius cycle .

Exercise 23 Let σ_c and σ_r be two metric signs and let a point (u₀,v₀)∈ℝ_p with v₀>0. Consider a cycle C_{σ_c}^s defined by

C_{σ_c}^s=(1,  u₀,  n,  u₀² −σ_r n²+є²), (13)

where

⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

v₀−

√

v₀²−(σ_r−σ_c)є²

σ_r−σ_c

if σ_r≠ σ_c,

є²

2v₀

if σ_r=σ_c.

(14)

Then,

The point (u₀, v₀) is σ_r-focus of the cycle.
The square of the σ_c-radius is exactly −є², i.e. (13) defines an infinitesimal-radius cycle.
The focal length of the cycle is an infinitesimal number of order є².

Hint: Combining two quadratic equations (one defines the squared σ_c-radius, another—v-coordinate of the focus), we found that n satisfies the equation:

Moreover, only the root from (14) of the quadratic case σ_r−σ_c≠ 0 gives an infinitesimal focal length. Then, we can find the m component of the cycle. The answer is also supported by CAS calculations. ⋄

The graph of cycle (13) in the parabolic space drawn at the scale of real numbers looks like a vertical ray starting at its focus, see Fig. 7.1(a), due to the following result.

Exercise 24 The infinitesimal cycle (13) consists of points which are infinitesimally close (in the sense of length from focus (7)) to its focus F=(u₀, v₀):

(u₀+є u, v₀+v₀u²+((σ_c−σ_r) u²−σ_r)

є²

4v₀

+O(є³)). (15)

Note that points below F (in the ordinary scale) are not infinitesimally close to F in the sense of length (7), but are in the sense of distance (3). Thus, having the set of points on the infinitesimal distance from an unknown point F we are not able to identify F. However, this is possible from the set of points on the infinitesimal length from F. Figure 7.1(a) shows elliptic, hyperbolic concentric and parabolic confocal cycles of decreasing radii which shrink to the corresponding infinitesimal-radius cycles.

It is easy to see that infinitesimal-radius cycles have properties similar to zero-radius ones, cf. Lemma 20.

Exercise 25 The image of SL₂(ℝ)-action on an infinitesimal-radius cycle (13) by conjugation (7) is an infinitesimal-radius cycle of the same order. The image of an infinitesimal cycle under cycle conjugation is an infinitesimal cycle of the same or lesser order.

Consideration of infinitesimal numbers in the elliptic and hyperbolic cases should not bring any advantages since the (leading) quadratic terms in these cases are non-zero. However, non-Archimedean numbers in the parabolic case provide a more intuitive and efficient presentation. For example, zero-radius cycles are not helpful for the parabolic Cayley transform (see Section 10.3), but infinitesimal cycles are their successful replacements. Another illustration is the second part of the following result as a useful substitution for Exercise 4.

Exercise 26 Let C_{σ_c}^s be the infinitesimal cycle (13) and G _{σ_c}^s=(k,l,n,m) be a generic cycle. Then:

Both the orthogonality condition C_{σ_c}^s⊥ G _{σ_c}^s (1) and the f-orthogonality G _{σ_c}^s ⊣ C_{σ_c}^s (15) are given by
ku₀²−2lu₀+m=O(є).

In other words, the cycle G _{σ_c}^s has the real root u₀.
The f-orthogonality (15) C_{σ_c}^s ⊣ G _{σ_c}^s is given by
ku₀²−2lu₀−2nv₀+m=O(є). (16)

In other words, the cycle G _{σ_c}^s passes the focus (u₀,v₀) of the infinitesimal cycle in the p-implementation.

It is interesting to note that the exotic f-orthogonality became a matching replacement for the usual one for infinitesimal cycles. Unfortunately, f-orthogonality is more fragile. For example, it is not invariant under a generic cycle conjugation (Exercise 38) and, consequently, we cannot use an infinitesimal-radius cycle to define a new parabolic inversion besides those shown in Fig. 6.4(c).

7.6 Infinitesimal Conformality

An intuitive idea of conformal maps, which is often provided in complex analysis textbooks for illustration purposes, is that they “send small circles into small circles with respective centres”. Using infinitesimal cycles, one can turn it into a precise definition.

Definition 27 A map of a region of ℝ_σ to another region is l-infinitesimally conformal for a length l (in the sense of Definition 9) if, for any l-infinitesimal cycle,

its image is an l-infinitesimal cycle of the same order and
the image of its centre or focus is displaced from the centre or focus of its image by an infinitesimal number of a greater order than its radius.

Remark 28 Note that, in comparison with Definition 13, we now work “in the opposite direction”. Formerly, we had the fixed group of motions and looked for corresponding conformal lengths and distances. Now, we take a distance or length (encoded in the infinitesimally-equidistant cycle) and check which motions respect it.

Natural conformalities for lengths from the centre in the elliptic and parabolic cases are already well studied. Thus, we are mostly interested here in conformality in the parabolic case, where lengths from the focus are more relevant. The image of an infinitesimal cycle (13) under SL₂(ℝ)-action is a cycle. Moreover, it is again an infinitesimal cycle of the same order by Exercise 25. This provides the first condition of Definition 27. The second part fulfils as well.

Exercise 29 Let G _{σ_c}^s be the image under g∈SL₂(ℝ) of an infinitesimal cycle C_{σ_c}^s from (13). Then, the σ_r-focus of G _{σ_c}^s is displaced from g(u₀,v₀) by infinitesimals of order є², while both cycles have σ_c-radius of order є.

Consequently, SL₂(ℝ)-action is infinitesimally-conformal in the sense of Definition 27 with respect to the length from the focus (Definition 9) for all combinations of σ, σ_c and σ_r.

Infinitesimal conformality seems to be intuitively close to Definition 13. Thus, it is desirable to understand a reason for the absence of exclusion clauses in Exercise 29 in comparison with Exercise 3.

Exercise 30 Show that, for lengths from foci (7) and σ_r= 0, the limit (10) at point y₀=u₀+ι v₀ does exist but depends on the direction y=u+ι v:

lim

t→ 0

d(g· y₀, g·(y₀+ty))

d(y₀, y₀+ty)

(d+cu₀)²+σ c² v₀² −2 K c v₀(d+c u₀)

, (17)

where K=u/v and g= (

a	b
c	d

). Thus, the length is not conformal.

However, if we consider points (15) of the infinitesimal cycle, then K=є u/v₀ u²= є/v₀ u. Thus, the value of the limit (17) at the infinitesimal scale is independent of y=u+ι v. It also coincides (up to an infinitesimal number) with the value in (27), which is defined through a different conformal condition.

Infinitesimal cycles are also a convenient tool for calculations of invariant measures, Jacobians, etc.

Remark 31 There are further connections between the infinitesimal number є and the dual unit ε. Indeed, in non-standard analysis , є² is a higher-order infinitesimal than є and can effectively be treated as 0 at the infinitesimal scale of є. Thus, it is simply a more relaxed version of the defining property of the dual unit ε²=0. This explains why many results of differential calculus can be deduced naturally within a dual numbers framework [58], which naturally absorbs many proofs from non-standard analysis.

Using this analogy between є and ε, we can think about the parabolic point space ℝ_p as a model for a subset of hyperreal numbers ℝ^* having the representation x+є y, with x and y being real. Then, a vertical line in ℝ_p (a special line, in Yaglom’s terms [339]) represents a monad , that is, the equivalence class of hyperreals, which are different by an infinitesimal number. Then, a Möbius transformation of ℝ_p is an analytic extension of the Möbius map from the real line to the subset of hyperreals.

The graph of a parabola is a section, that is, a “smooth” choice of a hyperreal representative from each monad. Geometric properties of parabolas studied in this work correspond to results about such choices of representatives and their invariants under Möbius transformations. It will be interesting to push this analogy further and look for a flow of ideas in the opposite direction: from non-standard analysis to parabolic geometry.

This concept is also known as Birkhoff orthogonality [39].

Lecture 7 Metric Invariants in Upper Half-Planes