Central Conics in H² are Fibers over the Group of Steiner Conics

1. Introduction

An intrinsic construction of conics in the real hyperbolic plane ${\mathbb{H}}^2$ would require only the incidence properties of the plane, the action of the collineation group, and a conformal metric. In this article, we provide such a construction by viewing ${\mathbb{H}}^2$ as a linear incidence geometry - two distinct points are on exactly one line and two distinct lines meet in at most one point. A first step toward such a construction was taken in [1] where we classified the Steiner conics in ${\mathbb{H}}^2$, defining them in the manner of [2] (pp. 134-140).

Definition 1.

Let T be a collineation of a planar incidence geometry. The Steiner conic  $E\left(T;P\right)$ afforded by T at point P is the locus of intersections $L\cap T\left(L\right)$ for the lines L concurrent at P. The conic $E\left(T_1;P_1\right)$ is congruent to $E\left(T;P\right)$ if, for some collineation $T_0$, $T_1=T_{0}T{T}_0^{-1}$ and $P_1=T_0\left(P\right)$. If the geometry is oriented, $E\left(T;P\right)$ is direct if T preserves orientation, opposite if T reverses orientation. The conic is degenerate if $T\left(P\right)=P$ or if $T\left(L\right)=L$ for some L.

It is possible for all conics in a planar geometry, defined algebraically or by other means, to be Steiner conics. This is the case in a projective plane over a field, as discussed in [3] (p. 80). From this viewpoint, the “conic sections” - ellipses, parabolas, hyperbolas - are classified by the invariants of affine collineations that afford them. By contrast, the collineation group of ${\mathbb{H}}^2$ consists entirely of isometries if a metric is imposed. This constrains the Steiner conics to the $E\left(T;P\right)$ in [1] where we showed them to be proper subsets of seven of the eleven categories of curves described in the classical sources [4,5]. Those categories extended work by Enrico D’Ovidio and others in the 1890s. They were described, non-intrinsically, by intersecting cones with a hyperbolic domain (analogous to the conic sections of Apollonius). The numbering and descriptors for these categories in [5] (p. 119), which correspond to the figures in [4] (p. 229), are reprised in Section 2 for the central conics, characterized by having two perpendicular lines of symmetry.

Since these are our subject, $E\left(T;P\right)$ will always denote a central Steiner conic, and C will always designate a central conic in ${\mathbb{H}}^2$. We write $\mathcal{D}$ for the set of direct $E\left(T;P\right)$ and $\mathcal{O}$ for the set of opposite $E\left(T;P\right)$. The set ${\mathcal{D}}_0$ consists of the members of $\mathcal{D}$ with no absolute points (on the boundary of ${\mathbb{H}}^2$ ). Upon choosing a set of congruence representatives ${\mathcal{D}}_0$ we use elliptic curve addition on their orthogonal trajectories to define the composition C of any two members of ${\mathcal{D}}_0$. This is our main result, which we state in Section 2. Apart from clarifying references to [1], our aim is to keep this presentation as self-contained as possible and of interest to the pedagogical community. The following outline provides context for the proof of our Main Theorem and its consequences in later sections.

In Section 3 we use the hyperboloid model of ${\mathbb{H}}^2$ to construct a family of elliptic space curves and describe the addition on their points. In Section 4 we represent points as complex numbers by projecting them into the conformal disk $\mathbb{D}$, where it is easy to visualize these curves and those in ${\mathcal{D}}_0$ as an orthogonal system. That the composition of a member of ${\mathcal{D}}_0$ with itself is a circle is proved in Section 5, followed by a formula in $\mathbb{D}$ for the elliptic curve addition. In Section 6 we derive some algebraic identities required for the proof, in Section 7, of the Main Theorem. In Section 8 the remaining categories of central conics listed in Section 2 will be produced from the compositions C. In Section 9 we use the Main Theorem and an intrinsic involution on the half-planes of ${\mathbb{H}}^2$ that interchanges ${\mathcal{D}}_0$ and $\mathcal{O}$ to describe a group structure on ${\mathcal{D}}_0\cup \mathcal{O}$. We define the fiber over each member of this group in Section 10 and partition the central conics into fibers.

For brevity throughout, we omit the details of routine calculations unless they are essential to the argument at hand. Several of these calculations are characteristically intricate. They have been performed manually and then thoroughly checked with the algebra program in Scientific Workplace 5.5.

2. Background and Statement of Main Theorem

Before stating our main result we establish some basic terminology by way of a brief summary of the classification of the $E\left(T;P\right)$. It is convenient to describe the translation component of T in terms of distance, whereby the eccentricity  $ϵ$ of $E\left(T;P\right)$ is a function of the distance s between P and $T\left(P\right)$. If $E\left(T;P\right)\in \mathcal{D}$ then $ϵ=sinhs$. We write $E_{ϵ}$ for the Steiner conic in this case and will work with a set of congruence representatives parameterized by $ϵ$. These representatives have common axes of symmetry, hence common center O. We reserve the term ellipse for an $E_{ϵ}$ with $0\le ϵ<1$, where $E_0$ is the degenerate ellipse O. The ellipses comprise ${\mathcal{D}}_0$. Each ellipse is a special case of a locus $C\left(\kappa ,\eta \right)$ consisting of points the sum of whose distances from two foci is $\kappa$, where $\eta$ is the distance between the foci; for an ellipse, $ϵ=tanh\kappa$ and $tanh\eta ={tanh}^2\kappa$. The center of $C\left(\kappa ,\eta \right)$ is the midpoint of the segment between the foci, its focal axisis the line through the foci, and its transverse axis is the line perpendicular to the focal axis at the center. If $ϵ>1$ we call $E_{ϵ}$ a hyper-ellipse. The conic $E_1$ is the pair of equidistant curves with angle-of-parallelism ${\displaystyle \frac{\pi }{4}}$ relative to the focal axis of the ellipses. The members of $\mathcal{O}$ are all central; we refer to them as hyperbolas and provide parameters for them in Section 9.

It will suffice to state the Main Theorem in terms of the ellipses because all of the $E\left(T;P\right)$ are related to them by the process of split-inversion. Split-inversion is a conformal, incidence-preserving involution on the open half-spaces of a hyperbolic space that interchanges complementary angles-of-parallelism. (The term is not standard but varies with context. See [6], where it is used without an explicit name.) In ${\mathbb{H}}^2$, we have the following definition.

Definition 2.

Let $\mathcal{P}$ be an open half-plane for a given line L in ${\mathbb{H}}^2$. For any $P\in \mathcal{P}$, let $L_P$ be the perpendicular to L through P. The split-inversion ofPinL is the point $P^{‡}\in L_P\cap \mathcal{P}$ such that the angle-of-parallelism to L at $P^{‡}$ is the complement of the angle-of-parallelism to L at P.

For example, if $ϵ\ne 0$ then split-inversion in the focal axis of the ellipse $E_{ϵ}$ interchanges it with the hyper-ellipse $E_{{\displaystyle \frac{1}{ϵ}}}$. In any inversive model of ${\mathbb{H}}^2$ (such as the Poincaré disk or upper half-plane) this is implemented by inversion in $E_1$, which sends each point on the axis to two absolute points. In particular, the split-inversion of $E_0$ is not in the plane, but it will be used in Section 9 to represent reflection in the transverse axis.

The central conics depicted in [4,5] comprise five of the categories in these sources, where they are designated as follows:

C1 Convex hyperbolas

C2 Concave hyperbolas

C4 Ellipses

C9 Equidistant curves

C10 Proper circles.

Since they fit our definition, we will include two other categories, afforded by degenerate cones:

C9A Pairs of intersecting lines

C9B Pairs of ultra-parallel lines.

Except for C9, C9A and C9B, these conics are represented in any inversive model of ${\mathbb{H}}^2$ by irreducible algebraic curves. We will see that the $C\left(\kappa ,\eta \right)$ account for C4 and C10. Split-inversion of the $C\left(\kappa ,\eta \right)$ in their focal axes will account for C2, and split-inversion of the $C\left(\kappa ,\eta \right)$ in their transverse axes for C1. Our construction will also produce the reducible curves C9, C9A and C9B.

The essential fact (see Section 4) is that the $E_{ϵ}$ are orthogonal in ${\mathbb{H}}^2$ to curves $F_{\alpha }$ defined by the following locus condition:

Let O be a designated point on a given line $L^{\prime }$ in ${\mathbb{H}}^2$. For $\alpha \in \left(-{\displaystyle \frac{\pi }{2}},0\right)\cup \left(0,{\displaystyle \frac{\pi }{2}}\right)$, let $F_{\alpha }$ be the locus of points P such that $\measuredangle OPA=\alpha$, where A is the foot of the perpendicular to $L^{\prime }$ from P.

Each $F_{\alpha }$ is symmetric about O. Reflection in $L^{\prime }$ reverses the orientation of $\angle OPA$, interchanging $F_{\alpha }$ and $F_{-\alpha }$. Our representatives for the $E_{ϵ}$ are the $E\left(T;P\right)$ for which O is the midpoint on $L^{\prime }$ of P and $T\left(P\right)$. For any $ϵ>0$, $L^{\prime }$ is considered to be the transverse axis of $E_{ϵ}$, and the line L through O perpendicular to $L^{\prime }$ is its focal axis. Each $F_{\alpha }$ meets $E_{ϵ}$ at two points in ${\mathbb{H}}^2$, one in each half-plane of the focal axis. The $F_{\alpha }$ are invariant under split-inversion in L. We will see in Section 3 that the $F_{\alpha }$ have a natural representation on the Minkowski hyperboloid. Using this model (and later its projection into the disk $\mathbb{D}$ ) we will derive the elliptic curve addition $P_1{\oplus }_{\alpha }{P}_2$ on the points of the algebraic curve containing $F_{\alpha }$. Our main result, the construction of conics in categories C4 and C10, is as follows.

Main Theorem.

Given ellipses  $E_{{ϵ}_1}$  and  $E_{{ϵ}_2}$  with  $0\le {ϵ}_1\le {ϵ}_2$  , let  $P_1$  and  $P_2$  be respective intersections with a given  $F_{\alpha }$  . Let  $\iota =1$  if  $P_1$  and  $P_2$  are chosen in the same half-plane of the focal axisLand let  $\iota =-1$  if they are chosen in opposite half-planes ofL.Then, with  ${ϵ}_1=tanh{\kappa }_1$  and  ${ϵ}_2=tanh{\kappa }_2$  , the set of points  $\left\{P_1{\oplus }_{\alpha }{P}_2\mid \alpha \in \left(-{\displaystyle \frac{\pi }{2}},0\right)\cup \left(0,{\displaystyle \frac{\pi }{2}}\right)\right\}$  for a consistent choice of  $\iota$  is the locus  $C_{\iota }\left(\kappa ,\eta \right)$  where  $\kappa ={\kappa }_2+\iota {\kappa }_1$  and  $\eta ={\eta }_2-{\eta }_1$.

We refer to $C_{\iota }\left(\kappa ,\eta \right)$ as the composition of $E_{{ϵ}_1}$ and $E_{{ϵ}_2}$ for the value $\iota$, denoting it as convenient by $E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}$. The intersections of $C_{\iota }\left(\kappa ,\eta \right)$ with L and $L^{\prime }$ are obtained as limiting cases for $\alpha$. Note in particular that if ${ϵ}_1=0$ then $C_{\iota }\left(\kappa ,\eta \right)=E_{{ϵ}_2}$. Also, if ${ϵ}_1={ϵ}_2$ then $\eta =0$, that is, the composition of an ellipse with itself is a circle ( $E_0$ if $\iota =-1$ ); we will prove this special case in advance of the general proof of the theorem. Note that $tanh\eta =tanh\left({\kappa }_2+{\kappa }_1\right)tanh\left({\kappa }_2-{\kappa }_1\right)$, from which it will follow as a corollary to the theorem that $C_{\iota }\left(\kappa ,\eta \right)$ is the composition of a unique pair of ellipses.

3. Constant-Angle Curves on the Minkowski Hyperboloid

The intrinsic view of non-Euclidean geometry promoted by Felix Klein was accelerated in the `pre-modern’ era by the introduction of the hyperboloid model of ${\mathbb{H}}^2$ in ${\mathbb{R}}^3$. Recall that this model is obtained from the quadric $\mathcal{H}$,

$$Z^2-X^2-Y^2=1$$

whose two sheets are ${\mathcal{H}}^{-}$, with $Z\le -1$, and the Minkowski hyperboloid  ${\mathcal{H}}^{+}$, with $Z\ge 1$. Let $N=\left(0,0,1\right)$ and $S=\left(0,0,-1\right)$. The points in this model are those on ${\mathcal{H}}^{+}$, and the lines are the intersections with ${\mathcal{H}}^{+}$ of planes through $\left(0,0,0\right)$. Each line, then, is identified with a normal vector $\mathbf{v}$ $=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$, where $v_1^2+v_2^2-v_3^2>0$, that determines its plane. The Minkowski inner product of two lines $\mathbf{v}$ and $\mathbf{w}$ is

$$〈\mathbf{v},\mathbf{w}〉=v_1{w}_1+v_2{w}_2-v_3{w}_3.$$

If $\mathbf{v}$ and $\mathbf{w}$ intersect, the angle $\theta$ between them in ${\mathcal{H}}^{+}$ is defined by

$$cos\theta ={\displaystyle \frac{〈\mathbf{v},\mathbf{w}〉}{\sqrt{〈\mathbf{v},\mathbf{v}〉}\sqrt{〈\mathbf{w},\mathbf{w}〉}}}.$$

Given $\alpha \in \left(-{\displaystyle \frac{\pi }{2}},0\right)\cup \left(0,{\displaystyle \frac{\pi }{2}}\right)$, let $F_{\alpha }$ be the intersection of $\mathcal{H}$ with the quadric $Xcot\alpha =YZ$. We will show that this space curve, which includes points on ${\mathcal{H}}^{-}$, is the complete form of the locus $F_{\alpha }$ defined in Section 2 (with $O=N$ ). First, for any point P on $F_{\alpha }$, let $P^Z$ be the reflection of P in the Z-axis. Then $P^Z$ is also on $F_{\alpha }$. Reflection in any of the coordinate planes interchanges $F_{\alpha }$ and $F_{-\alpha }$, so we work with $0<\alpha <{\displaystyle \frac{\pi }{2}}$ in this section. For $\varphi \in [\alpha -{\displaystyle \frac{\pi }{2}},0)\cup (0,{\displaystyle \frac{\pi }{2}}-\alpha ]$, let

$$Q_{\alpha }\left(\varphi \right)={\displaystyle \frac{1}{\sqrt{2}cos\alpha }}\sqrt{cos2\alpha +cos2\varphi }.$$

A one-to-one parameterization of $F_{\alpha }$ is given by

$$\begin{array}{cc}\hfill X& =\pm Q_{\alpha }\left(\varphi \right)cot\alpha cot\varphi \hfill \\ \hfill Y& =\pm Q_{\alpha }\left(\varphi \right)cot\alpha \hfill \\ \hfill Z& =cot\alpha cot\varphi \hfill \end{array}$$

(1)

whereby $\varphi ={\displaystyle \frac{\pi }{2}}-\alpha$ corresponds to N, and $\varphi =\alpha -{\displaystyle \frac{\pi }{2}}$ to S. Now let P be any point on $F_{\alpha }\cap {\mathcal{H}}^{+}$, so $\varphi \in (0,{\displaystyle \frac{\pi }{2}}-\alpha ]$, and let $L^{\prime }$ be the line $\mathbf{v}$ $=\mathbf{j}$. The line through N and P is ${\mathbf{v}}_1$ $=\left(sin\varphi \right)\mathbf{i}-\left(cos\varphi \right)$ $\mathbf{j}$ and the line through P orthogonal to $L^{\prime }$ is ${\mathbf{v}}_2$ $=\mathbf{i}-Q_{\alpha }\left(\varphi \right)\mathbf{k}$, so the cosine of the angle between ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ is

$${\displaystyle \frac{sin\varphi }{\sqrt{1-Q_{\alpha }^2\left(\varphi \right)}}}=cos\alpha .$$

Thus, the angle between the line through N and P and the line through P orthogonal to $L^{\prime }$ is $\alpha$.

A complete $F_{\alpha }$ is an example of a real elliptic curve obtained from the intersection of two quadrics. As such, it has an addition structure, with identity N, and $P^Z$ the inverse of P. We are particularly interested in the sum $P_1{\oplus }_{\alpha }{P}_2$ when $P_1$ and $P_2$ are on $F_{\alpha }\cap {\mathcal{H}}^{+}$. This sum is constructed from the the S-line through these points, the intersection of ${\mathcal{H}}^{+}$ with a plane through S. Relative to the Minkowski distance formula (see Section 4), the dilation with hyperbolic factor 2 and center N transforms lines into S-lines. Now, the S-line through $P_1$ and $P_2$ (the S-line tangent to $F_{\alpha }$ if $P_1=P_2$ ) intersects $F_{\alpha }$ at one other point R. Then $P_1{\oplus }_{\alpha }{P}_2=R^Z$. We have $P{\oplus }_{\alpha }N=P$ for any P on $F_{\alpha }$ because the S-line through P and N is $\mathbf{v}$ $=\mathbf{i}-\left(cot\varphi \right)\mathbf{j}$, so $R=P^Z$ in this case. In general, it is not difficult to obtain the addition formula explicitly. From (1), $Y^2={\displaystyle \frac{Z^2-1}{Z^2{tan}^2\alpha +1}}$, so it suffices to determine the Z-coordinate of $P_1{\oplus }_{\alpha }{P}_2$. The vector calculations are straightforward, though characteristically tedious. Later we will derive an alternative formula for addition on $F_{\alpha }$ from the stereographic projection of $\mathcal{H}$ into ${\mathbb{R}}^2$, but we note here that the Z-coordinate of $P{\oplus }_{\alpha }P$ takes a particularly simple form that we will use in Section 5: If P is on  ${\mathcal{H}}^{+}$  with  $Y\ge 0$  then the Z-coordinate of  $P{\oplus }_{\alpha }P$  is

$${\displaystyle \frac{{cos}^2\alpha -{sin}^4\varphi }{{sin}^2\alpha -{cos}^4\varphi }}.$$

(2)

Since $Q_{\alpha }\left(\varphi \right)$ is real, it follows from (2) that $R^Z=P{\oplus }_{\alpha }P$ is on ${\mathcal{H}}^{+}$ provided ${cos}^2\varphi <sin\alpha$. On the other hand, Figure 1 shows an example where R (and therefore $R^Z$ ) is on ${\mathcal{H}}^{-}$.

Generally, if $P_1$ and $P_2$ are distinct points on ${\mathcal{H}}^{+}$ we will see (Section 5) that $R^Z$ is on ${\mathcal{H}}^{+}$ provided $cos{\varphi }_{1}cos{\varphi }_2<sin\alpha$; it is sufficient, but not necessary, that $P_1$ and $P_2$ be in the open region $\mathcal{E}$ where $-1<X<1$ for this condition to hold. In any case, our constructions will take place within ${\mathcal{H}}^{+}$. Figure 2 is an illustration on ${\mathcal{H}}^{+}$ of the Main Theorem. The ellipses $E_{{\displaystyle \frac{1}{3}}}$ and $E_{{\displaystyle \frac{2}{3}}}$ are shown in red, with $E_{{\displaystyle \frac{1}{3}}}$ closer to $N=E_0$. Their compositions C for $\iota =\pm 1$ are shown in black. When $\iota =-1$ the conic C intersects $E_{{\displaystyle \frac{1}{3}}}$ in four points.

4. Projection to Plane Curves

Stereographic projection of $\mathcal{H}$, from S into ${\mathbb{R}}^2$, by

$$\left(X,Y,Z\right)\mapsto \left({\displaystyle \frac{X}{1+Z}},{\displaystyle \frac{Y}{1+Z}}\right)=\left(x,y\right)$$

conformally maps ${\mathcal{H}}^{+}$ to the interior of $\mathbb{D}$ and ${\mathcal{H}}^{-}$ to the region outside the disk. Thus, N is mapped to $O=\left(0,0\right)$, and $L^{\prime }$ becomes the line $y=0$ in $\mathbb{D}$. From (1), we find that the image of $F_{\alpha }$ is the polar curve

$$r^2={\displaystyle \frac{cos\left(\alpha +\varphi \right)}{cos\left(\alpha -\varphi \right)}}.$$

(3)

This is the non-singular cubic $F_{\alpha }\left(x,y\right)=0$, where

$$F_{\alpha }\left(x,y\right)=\left(x^2+y^2\right)\left(x+ytan\alpha \right)-\left(x-ytan\alpha \right)$$

(3a)

which has the single real asymptote $xcot\alpha +y=0$. The portion in $\mathbb{D}$ of an $F_{\alpha }$ is shown in blue in Figure 3, with constant angle $\alpha$ at P.

This representation of $F_{\alpha }$ as a cubic curve symmetric about O was chosen to make computation as simple as possible. However, our results are not representation dependent because the image of $F_{\alpha }$ by a Möbius transformation is either a cubic or an irreducible quartic with two ordinary singularities at non-real points. In particular, incidence is preserved and the elliptic curve addition law we derive would hold in either case.

We now show that the $F_{\alpha }$ and $E_{ϵ}$ are orthogonal trajectories of each other. Since stereographic projection is conformal, it follows from (3) that the orthogonal trajectories of the $F_{\alpha }$ are the solutions to the polar equation

$$\left(sin2\varphi \right)d\varphi ={\displaystyle \frac{1-r^4}{2r^3}}dr.$$

These are the algebraic curves

$$r^4-2r^2\left(cos2\varphi +{\displaystyle \frac{2}{{ϵ}^2}}\right)+1=0$$

(4)

with parameter $ϵ$, which are the complete forms of the non-trivial $E_{ϵ}$ [1] (pp. 139-140). Figure 4 shows the portions in $\mathbb{D}$ of $F_{\alpha }$ (blue) and $E_{ϵ}$ (red). The curve $E_1$, shown in bold red, is the projection of the intersections with ${\mathcal{H}}^{+}$ of the planes $X=\pm 1$, the boundary of the region $\mathcal{E}$ defined in Section 2. The ellipses are in $\mathcal{E}$.

We now identify $P=\left(x,y\right)$ with the complex number $x+iy$. The image of the transverse axis $L^{\prime }$ is the real line in $\mathbb{D}$ and the image of the focal axis L is the imaginary line. Writing $P^{*}$ for $x-iy$, the distance between two points $P_1$ and $P_2$ in $\mathbb{D}$ is given by

$$d\left(P_1,P_2\right)=arctanh\left|{\displaystyle \frac{P_1-P_2}{1-P_1^{*}{P}_2}}\right|$$

(5)

which is equivalent to the Minkowski distance formula on ${\mathcal{H}}^{+}$ given by

$${\displaystyle \frac{1}{2}}arccosh\left(Z_1{Z}_2-Y_1{Y}_2-X_1{X}_2\right).$$

Also, the complete curves $F_{\alpha }$ and $E_{ϵ}$ are invariant under the transformation $P\mapsto {\displaystyle \frac{1}{P}}$. For $F_{\alpha }$, this is equivalent to replacing $\varphi$ with $-\varphi$ in (1), whereby ${\displaystyle \frac{1}{0}}$ is the ideal point for the real asymptote, with projective coordinates $\left[tan\alpha :-1:0\right]$. We note that the split-inversion of P in L is

$${\displaystyle \frac{1\pm P^{*}}{P^{*}\mp 1}}$$

with the signs determined by the half-plane containing P. The split-inversion of P in $L^{\prime }$ is

$${\displaystyle \frac{1\pm i{P}^{*}}{P^{*}\pm i}}.$$

Remark 1.

The Möbius involution $P\mapsto {\displaystyle \frac{P-i}{iP-1}}$ maps $\mathbb{D}$ to the upper half-plane, its boundary to the real line, and the remaining points to the lower half-plane. The $F_{\alpha }$ are mapped to the one-loop Cassini ovals and the $E_{ϵ}$ to the two-loop ovals. From the algebraic classification of Cassini ovals it follows that the $F_{\alpha }$ and $E_{ϵ}$ represent all genus 1 curves with real coefficients, which are algebraically equivalent precisely when they have the same shape invariant, Klein’s j-invariant. The invariant j can be computed from the cross-ratio $\chi$ of an algebraically equivalent non-singular cubic; the curve is harmonic if $j=1$, suitably normalized. For $F_{\alpha }$ we can take $\chi =e^{4i\alpha }$ which implies $j\le 1$; for $E_{ϵ}$ we can take $\chi ={ϵ}^4$ which implies $j\ge 1$. Thus, there is no overlap by algebraic equivalence between the $F_{\alpha }$ and $E_{ϵ}$ except for the harmonic cases ( $\alpha =\pm {\displaystyle \frac{\pi }{4}}$, $ϵ=2^{\pm {\displaystyle \frac{1}{4}}}$ ).

5. Composition of Ellipsesp

In this section we derive an explicit formula for the elliptic curve addition on $F_{\alpha }$ with points represented as complex numbers. First, we recall the simple form of (2) in Section 3 and use it to prove a special case of the Main Theorem that motivates the composition of two ellipses in general. Specifically, with ${cos}^2\varphi <sin\alpha$, it is easy to find the composition of $E_{ϵ}$ with itself. We assume $\iota =1$ since the symmetry of $F_{\alpha }$ implies $E_{ϵ}{\circ }_{-1}{E}_{ϵ}=E_0=O$.

Proposition 1.

The composition  $E_{ϵ}{\circ }_1{E}_{ϵ}$  is the circle with center O and radius  $arctanhϵ$.

Proof.

Let P be an intersection point of $F_{\alpha }$ with $E_{ϵ}$. Solving (3) and (4) simultaneously, with ${sin}^2\alpha \le {cos}^2\varphi <sin\alpha$ because $E_{ϵ}\subset \mathcal{E}$, yields

$${ϵ}^2=2{\displaystyle \frac{cos2\alpha +cos2\varphi }{{sin}^{2}2\varphi }}.$$

(6)

Thus, from (2) and (6), the Z-coordinate of $P{\oplus }_{\alpha }P$ on ${\mathcal{H}}^{+}$ is

$$Z={\displaystyle \frac{{cos}^2\alpha -{sin}^4\varphi }{{sin}^2\alpha -{cos}^4\varphi }}={\displaystyle \frac{1+{ϵ}^2}{1-{ϵ}^2}}.$$

Since $X^2+Y^2=Z^2-1$, it follows that

$$x^2+y^2={\left({\displaystyle \frac{X}{1+Z}}\right)}^2+{\left({\displaystyle \frac{Y}{1+Z}}\right)}^2={ϵ}^2,$$

the circle in $\mathbb{D}$ with center O and radius $arctanhϵ$. □

If $ϵ\ne 0$ then the vertices of $E_{ϵ}$ on its focal axis in $\mathbb{D}$ are $\pm {\displaystyle \frac{1}{ϵ}}\left(\sqrt{1-{ϵ}^2}-1\right)i$. It follows from Formula (5) that the radius of the circle in Proposition 1 is the length of the segment between these vertices. Consequently, any circle in ${\mathbb{H}}^2$ is the composition of a unique ellipse with itself, but no non-trivial circle is an ellipse. We now proceed under the assumptions that ${ϵ}_2\ne {ϵ}_1$ and ${ϵ}_1{ϵ}_2\ne 0$.

Next, with $P=x+iy$ we obtain a formula for $R^Z=P_1{\oplus }_{\alpha }{P}_2$ when $P_1\ne P_2$. This formula applies to the complete $F_{\alpha }$. First, the projection into $\mathbb{D}$ of the S-line through $P_1$ and $P_2$ is a chord of the unit circle whose points P satisfy

$$\left(P_1-P_2\right)P^{*}-\left(P_1^{*}-P_2^{*}\right)P=P_1{P}_2^{*}-P_2{P}_1^{*}.$$

(7)

Since N projects to the identity 0 we assume $P_1{P}_2\ne 0$. Next, from (3a), $F_{\alpha }$ consists of all P such that

$$\left(P{P}^{*}-1\right)\left(P+P^{*}\right)=i\left(P{P}^{*}+1\right)\left(P-P^{*}\right)tan\alpha .$$

(8)

Solving for $P^{*}$ in (7) and substituting into (8) produces a cubic polynomial in P with constant term $a_0$. Since $P_1$ and $P_2$ are on $F_{\alpha }$, the roots of this polynomial are obtained from the monic factorization $\left(P-P_1\right)\left(P-P_2\right)\left(P-R\right)$. Then $P_1{P}_{2}R=-a_0$ and $P_1{\oplus }_{\alpha }{P}_2=-R$. It follows after solving for R that

$$P_1{\oplus }_{\alpha }{P}_2={\displaystyle \frac{P_1-P_2}{P_1^{*}-P_2^{*}}}\left({\displaystyle \frac{P_1^{*}}{P_1}}-{\displaystyle \frac{P_2^{*}}{P_2}}\right){\displaystyle \frac{1}{\left(P_1-P_2\right)+e^{2i\alpha }\left(P_1^{*}-P_2^{*}\right)}}.$$

(9)

Formula (9) is undefined when ${\displaystyle \frac{P_2-P_1}{P_1^{*}-P_2^{*}}}=e^{2i\alpha }$, but this happens only when (7) is parallel to the asymptote of $F_{\alpha }$, equivalently, when $P_1{P}_2=1$. In this case the sum is the ideal point $\left[tan\alpha :-1:0\right]$. Otherwise, from (3), let $P_j=r_j{e}^{i{\varphi }_j}$. Then ${\left|{\displaystyle \frac{P_1^{*}}{P_1}}-{\displaystyle \frac{P_2^{*}}{P_2}}\right|}^2=2\left(1-cos\left(2{\varphi }_1-2{\varphi }_2\right)\right)$. Thus ${\left|P_1{\oplus }_{\alpha }{P}_2\right|}^2$, which will be used in the proof of the Main Theorem, is determined by the denominator of the third factor in (9). Noting that $cos2\alpha +cos2{\varphi }_j\ge 0$, there are two cases to consider depending on whether $cos\left(\alpha -{\varphi }_1\right)$ and $cos\left(\alpha -{\varphi }_2\right)$ have the same or opposite signs. For brevity we omit the straightforward calculation that yields

$${\left|\left(P_1-P_2\right)+e^{2i\alpha }\left(P_1^{*}-P_2^{*}\right)\right|}^2=2{\left(\sqrt{cos2\alpha +cos2{\varphi }_1}-\iota \sqrt{cos2\alpha +cos2{\varphi }_2}\right)}^2,$$

where $\iota =1$ in the same-sign case, and $\iota =-1$ otherwise. Therefore, with $Q_{\alpha }\left(\varphi \right)$ as in Section 3,

$${\left|P_1{\oplus }_{\alpha }{P}_2\right|}^2={\displaystyle \frac{1}{cos2\alpha +1}}{\displaystyle \frac{1-cos\left(2{\varphi }_1-2{\varphi }_2\right)}{{\left(Q_{\alpha }\left({\varphi }_1\right)-\iota Q_{\alpha }\left({\varphi }_2\right)\right)}^2}}.$$

(10)

Proposition 2.

If  $P_1$  and  $P_2$  are on  $F_{\alpha }$  in  $\mathbb{D}$  then  $P_1{\oplus }_{\alpha }{P}_2$  is in  $\mathbb{D}$  if and only if  $cos{\varphi }_{1}cos{\varphi }_2<sin\alpha$.

Proof. 

Since $F_{\alpha }$ is symmetric about 0 it suffices to assume $P_1$ and $P_2$ are in the first polar quadrant of $\mathbb{D}$. The claim is obvious if $P_1$ and $P_2$ are in opposite quadrants because $cos{\varphi }_{1}cos{\varphi }_2<0$ in that case and $P_1{\oplus }_{\alpha }{P}_2$ is clearly in $\mathbb{D}$. Then $\iota =1$ in (10). Instead of a tedious manipulation with identities, we show that the condition for $\left|P_1{\oplus }_{\alpha }{P}_2\right|<1$ is determined by the location of $P_1$ and $P_2$ relative to the portion of $E_1$ in this quadrant. First, if $P_1$ and $P_2$ are related by inversion in $E_1$, then

$$P_2={\displaystyle \frac{1-P_1^{*}}{1+P_1^{*}}}$$

whereby

$$tan{\varphi }_2={\displaystyle \frac{2r_1}{1-r_1^2}}sin{\varphi }_1.$$

Then $tan\alpha tan{\varphi }_2=Q_{\alpha }\left({\varphi }_1\right)$, equivalently, $cos{\varphi }_{1}cos{\varphi }_2=sin\alpha$. In this case, the S-line through $P_1$ and $P_2$ meets the boundary at $R=-1$, as shown in Figure 5 for $P_1=P_2$ on $E_1$ and for $P_1\ne P_2$. Now, if $P_2$ is replaced by any point $P\left(\varphi \right)$ on $F_{\alpha }$ in the first quadrant then the corresponding R is in $\mathbb{D}$ if and only if ${\varphi }_2>\varphi$, that is, $cos\varphi cos{\varphi }_2<sin\alpha$. $\square$

In particular, if $P_1$ and $P_2$ are both in $\mathcal{E}$ then $P_1{\oplus }_{\alpha }{P}_2$ is in $\mathbb{D}$, consistent with Proposition 1 and the statement of the Main Theorem.

6. The Computation Field

Typical of analytic geometry in the hyperbolic plane, the proof of the Main Theorem requires several algebraic identities that are straightforward but intricate. We establish these in this section. Although the dilation that transforms lines to S-lines also transforms ellipses to quadratic curves in $\mathbb{D}$, it also transforms each $F_{\alpha }$ to a quartic curve and offers no advantage in the proof. To keep the algebra manageable, we view ${ϵ}_1$, ${ϵ}_2$, and $cos2\alpha$ as formal variables and work within an extension of the base field  $\mathbb{Q}\left({ϵ}_1,{ϵ}_2,cos2\alpha \right)$. For $j=1$ or 2, let ${\beta }_j={ϵ}_j^{2}cos2\alpha -1$ and ${\gamma }_j={ϵ}_j^4-2{\beta }_j-1$. Adjoining $\sqrt{{\gamma }_1}$ and $\sqrt{{\gamma }_2}$ to the base field produces an extension K with basis $B_0=1$, $B_1=\sqrt{{\gamma }_1}$, $B_2=\sqrt{{\gamma }_2}$, $B_3=B_1{B}_2$. Let $B_4=\sqrt{B_1+{\beta }_1}\sqrt{B_2+{\beta }_2}$. Then each $B_j\ge 0$ if $\alpha$ and ${ϵ}_j$ are real numbers with $0\le {ϵ}_j<1$. We will avoid explicit reference to the parameter $\alpha$ by performing computations in the field $K\left(B_4\right)$, where coefficients of the basis elements reduce to polynomials in ${\beta }_1$ and ${\beta }_2$ over the base field.

Certain elements in $\mathbb{Q}\left({ϵ}_1,{ϵ}_2\right)$ will occur frequently as we manipulate the field relations. These include

$$\begin{array}{cc}\hfill \delta & ={ϵ}_1^2-{ϵ}_2^2\hfill \\ \hfill \sigma & ={ϵ}_1^2+{ϵ}_2^2\hfill \\ \hfill \tau & =\iota {ϵ}_1{ϵ}_2\hfill \\ \hfill \omega & =\tau +{\displaystyle \frac{1}{2}}\sigma \hfill \\ \hfill \lambda & ={\tau }^2+1\hfill \\ \hfill \mu & ={\tau }^2-1.\hfill \end{array}$$

For example, the elements ${\beta }_0:=2\tau cos2\alpha +\lambda$ and ${\gamma }_0:=2{\tau }^{2}cos2\alpha -\sigma$ can be written as

$$\begin{array}{cc}\hfill {\beta }_0& =\lambda +{\displaystyle \frac{2\tau }{\delta }}\left({\beta }_1-{\beta }_2\right)\hfill \\ \hfill {\gamma }_0& ={ϵ}_2^2{\beta }_1+{ϵ}_1^2{\beta }_2.\hfill \end{array}$$

Also, the non-zero element $\delta$ can be expressed as ${\beta }_1{ϵ}_2^2-{\beta }_2{ϵ}_1^2$, so the element

$$\mathrm{\Omega }={ϵ}_1^2\left({\beta }_2+1\right)-{ϵ}_2^2\left({\beta }_1+1\right)$$

is zero. To prove the pivotal lemma that concludes this section, we will perform computations $mod\left(\mathrm{\Omega }\right)$, where $\left(\mathrm{\Omega }\right)$ consists of all multiples of $\mathrm{\Omega }$. For example,

$${\gamma }_0=\tau \left({\beta }_0-\mu \right)-2\omega$$

(11)

because subtracting the RHS from the LHS produces ${\displaystyle \frac{\sigma }{\delta }}\mathrm{\Omega }$. Similarly,

$$\tau {\gamma }_1{\gamma }_2={\beta }_0\left({\gamma }_0+\tau \mu \right)+2\omega \left(\tau {\gamma }_0-\mu \right)$$

(12)

because subtraction produces ${\displaystyle \frac{1}{\delta }}\left(2\tau \left({\beta }_1+{\beta }_2\right)+4{\tau }^3+\sigma \mu -{\sigma }^2\tau \right)\mathrm{\Omega }$.

Lemma 1.

Let $U_1=\mu +B_1+B_2-B_3$  and $U_2={\gamma }_0+{ϵ}_2^2{B}_1+{ϵ}_1^2{B}_2$ . Then

$$\left(\tau {\beta }_0-2\omega +\tau B_3\right)U_1=\left(2\tau \omega -{\beta }_0-B_3\right)U_2.$$

Proof. 

We have $B_3\left(b_0+b_1{B}_1+b_2{B}_2+b_3{B}_3\right)=b_3{\gamma }_1{\gamma }_2+b_2{\gamma }_2{B}_1+b_1{\gamma }_1{B}_2+b_0{B}_3$, so

$$\begin{array}{cc}\hfill \left(\tau {\beta }_0-2\omega +\tau B_3\right)U_1& =u_0+u_1{B}_1+u_2{B}_2+u_3{B}_3\hfill \\ \hfill \left(2\tau \omega -{\beta }_0-B_3\right)U_2& =v_0+v_1{B}_1+v_2{B}_2+v_3{B}_3\hfill \end{array}$$

with the coefficients of the $B_j$ as follows.

$u_0=\tau \left(\mu {\beta }_0-{\gamma }_1{\gamma }_2\right)-2\mu \omega$	$v_0=\left(2\tau \omega -{\beta }_0\right){\gamma }_0$
$u_1=\tau {\beta }_0+\tau {\gamma }_2-2\omega$	$v_1={ϵ}_2^2\left(2\tau \omega -{\beta }_0\right)-{ϵ}_1^2{\gamma }_2$
$u_2=\tau {\beta }_0+\tau {\gamma }_1-2\omega$	$v_2={ϵ}_1^2\left(2\tau \omega -{\beta }_0\right)-{ϵ}_2^2{\gamma }_1$
$u_3=\tau \left(\mu -{\beta }_0\right)+2\omega$	$v_3=-{\gamma }_0$

Then $u_3=v_3$ and $u_0=v_0$ by (11) and (12) respectively, whereas $\left(u_1-v_1\right)\delta =2{ϵ}_1\left({ϵ}_1+\iota {ϵ}_2\right)\mathrm{\Omega }$ and $\left(u_2-v_2\right)\delta =2{ϵ}_2\left({ϵ}_2+\iota {ϵ}_1\right)\mathrm{\Omega }$. $\square$

Corollary 1.

Let  $\rho ={\displaystyle \frac{U_1-2B_4}{U_2-2\tau B_4}}$. Then  ${\rho }^2-{\displaystyle \frac{{\beta }_0+B_3}{\omega }}\rho +1=0$.

Proof. 

From Lemma 1 we have ${\displaystyle \frac{{\beta }_0+B_3}{\omega }}=2{\displaystyle \frac{U_1+\tau U_2}{U_2+\tau U_1}}$. Then

$${\left(U_2-2\tau B_4\right)}^2\left({\rho }^2-2{\displaystyle \frac{U_1+\tau U_2}{U_2+\tau U_1}}\rho +1\right)={\displaystyle \frac{\tau U_1-U_2}{U_2+\tau U_1}}\left(U_1^2-U_2^2+4\mu B_4^2\right).$$

Now $B_4^2=\left({\beta }_1{\beta }_2+{\beta }_2{B}_1+{\beta }_1{B}_2\right)$, and

$$\begin{array}{cc}\hfill U_1^2& =\left({\mu }^2+{\gamma }_1{\gamma }_2+{\gamma }_1+{\gamma }_2\right)+2\left(\mu -{\gamma }_2\right)B_1+2\left(\mu -{\gamma }_1\right)B_2+2\left(1-\mu \right)B_3\hfill \\ \hfill U_2^2& =\left({\gamma }_0^2+{ϵ}_2^4{\gamma }_1+{ϵ}_1^4{\gamma }_2\right)+2{\gamma }_0{ϵ}_2^2{B}_1+2{\gamma }_0{ϵ}_1^2{B}_2+2{\tau }^2{B}_3\hfill \end{array}$$

whereby $U_1^2-U_2^2=c_0+2c_1{B}_1-2c_2{B}_2-4\mu B_3$ with

$$\begin{array}{cc}\hfill c_0& ={\delta }^2-\left({\beta }_1^2{ϵ}_2^4+{\beta }_2^2{ϵ}_1^4\right)-2\left(\mu -1\right){\beta }_1{\beta }_2\hfill \\ \hfill c_1& ={ϵ}_2^2\delta -{\beta }_1{ϵ}_2^4-\left(\mu -1\right){\beta }_2\hfill \\ \hfill c_2& ={ϵ}_1^2\delta +\left(\mu -1\right){\beta }_1+{\beta }_2{ϵ}_1^4.\hfill \end{array}$$

Since $\delta ={\beta }_1{ϵ}_2^2-{\beta }_2{ϵ}_1^2$, these coefficients reduce to $c_0=-4\mu {\beta }_1{\beta }_2$, $c_1=-2\mu {\beta }_2$, $c_2=2\mu {\beta }_1$. It follows that $U_1^2-U_2^2+4\mu B_4^2=0$. $\square$

7. Proof of the Main Theorem

In this section we assume $0\le {ϵ}_1\le {ϵ}_2$ and begin with an illustration in $\mathbb{D}$ of $C=C_{\iota }\left(\kappa ,\eta \right)$ for the two cases $\iota =\pm 1$. Figure 6 shows ellipses $E_{{ϵ}_1}$ and $E_{{ϵ}_2}$ (red) and S-lines through $P_1=E_{{ϵ}_1}\cap F_{\alpha }$ and $P_2=E_{{ϵ}_2}\cap F_{\alpha }$, for various $F_{\alpha }$ (blue) when $P_1$ and $P_2$ are in the same half-plane of the focal axis L. The third intersection with each $F_{\alpha }$ is shown on C (black) in the opposite half-plane, so $P_1{\oplus }_{\alpha }{P}_2$, the reflection in O of the third intersection, is also on C.

If $\iota =-1$ then C intersects $E_{{ϵ}_1}$ at four points, as shown in Figure 7 for the same two ellipses and a typical $F_{\alpha }$ when $P_1$ and $P_2$ are in opposite half-planes of L.

Remark 2.

By symmetry, C is produced without reflecting the third intersection with each $F_{\alpha }$ in O. We have followed the definition of elliptic curve addition to ensure associativity in Section 9 where the ellipses will be viewed as oriented figures with a group structure.

Proof of the MainTheorem.

Let $\pm itanh{\displaystyle \frac{\eta }{2}}$ be the foci of $C_{\iota }\left(\kappa ,\eta \right)$ in the theorem, and let $\epsilon ={\displaystyle \frac{tanh\eta }{tanh\kappa }}$. A direct calculation using (5) shows that $C_{\iota }\left(\kappa ,\eta \right)$ is described by the polar equation

$${\displaystyle \frac{1+r^4}{2r^2}}={\displaystyle \frac{2-{\epsilon }^2-{tanh}^2\kappa +\left({\epsilon }^2-{tanh}^2\eta \right)cos2\theta }{{tanh}^2\kappa -{tanh}^2\eta }}.$$

(13)

The strategy is as follows. The theorem asserts that

$$\begin{array}{cc}\hfill {tanh}^2\kappa & ={\displaystyle \frac{\sigma +2\tau }{\lambda +2\tau }}\hfill \\ \hfill {tanh}^2\eta & ={\displaystyle \frac{{\delta }^2}{{\mu }^2}}\hfill \end{array}$$

whereby (13) becomes

$$cos2\theta ={\displaystyle \frac{\omega r^4-2\lambda r^2+\omega }{\left(\sigma -2\tau \right)r^2}}.$$

(14)

Now let $\phi$ be the polar angle for $P_1{\oplus }_{\alpha }{P}_2$ and let $\rho ={\left|P_1{\oplus }_{\alpha }{P}_2\right|}^2$. Since $P_1$ and $P_2$ are on $F_{\alpha }$ it follows from (3), with $\rho ={\displaystyle \frac{cos\left(\alpha +\phi \right)}{cos\left(\alpha -\phi \right)}}$, that

$$cos2\phi ={\displaystyle \frac{\left(1+{\rho }^2\right)cos2\alpha -2\rho }{2\rho cos2\alpha -\left(1+{\rho }^2\right)}}.$$

(15)

We must show that $cos2\phi$ in (15) is equal to $cos2\theta$ in (14) when $r^2=\rho$. That is, since ${\beta }_0=\lambda +2\tau cos2\alpha$, we will show that $\rho$ is a root of the quartic polynomial

$$W^4-2a{W}^3+2\left(b+1\right)W^2-2aW+1$$

with $a={\displaystyle \frac{1}{\omega }}{\beta }_0$ and $b={\displaystyle \frac{1}{2{\omega }^2}}\left({\beta }_0^2-B_3^2\right)$. The roots occur in reciprocal pairs and the polynomial factors as

$$\left(W^2-{\displaystyle \frac{{\beta }_0+B_3}{\omega }}W+1\right)\left(W^2-{\displaystyle \frac{{\beta }_0-B_3}{\omega }}W+1\right).$$

We will show that $\rho$ is a root of the first factor.

First we write $\rho$ as an element of $K\left(B_4\right)$. Let $U_1$ and $U_2$ be as in Lemma 1. From (10),

$$\rho ={\displaystyle \frac{1}{2{cos}^2\alpha }}{\displaystyle \frac{1-cos\left(2{\varphi }_1-2{\varphi }_2\right)}{{\left(Q_{\alpha }\left({\varphi }_1\right)-Q_{\alpha }\left({\varphi }_2\right)\right)}^2}}$$

and from (6),

$$cos2{\varphi }_1={\displaystyle \frac{B_1-1}{{ϵ}_1^2}},cos2{\varphi }_2={\displaystyle \frac{B_2-1}{{ϵ}_2^2}}.$$

Then $sin2{\varphi }_j={\displaystyle \frac{\sqrt{2}}{{ϵ}_j^2}}\sqrt{{\beta }_j+B_j}$, so ${\tau }^{2}sin2{\varphi }_{1}sin2{\varphi }_2=2B_4$ and

$$1-cos\left(2{\varphi }_1-2{\varphi }_2\right)={\displaystyle \frac{1}{{\tau }^2}}\left(U_1-2B_4\right).$$

Further,

$$\sqrt{cos2\alpha +cos2{\varphi }_j}={\displaystyle \frac{1}{{ϵ}_j}}\sqrt{B_j+{\beta }_j}.$$

Recall ${\gamma }_0={\beta }_1{ϵ}_2^2+{\beta }_2{ϵ}_1^2$. Then

$${\left(Q_{\alpha }\left({\varphi }_1\right)-Q_{\alpha }\left({\varphi }_2\right)\right)}^2={\displaystyle \frac{1}{{\gamma }_0+\sigma }}\left(U_2-2\tau B_4\right).$$

Thus,

$$\rho ={\displaystyle \frac{U_1-2B_4}{U_2-2\tau B_4}}.$$

By Corollary 1, $\rho$ is a root of $W^2-{\displaystyle \frac{{\beta }_0+B_3}{\omega }}W+1$ as claimed. $\square$

Remark 3.

Though not needed, it can be shown that the roots of the second factor are conjugate points on the unit circle. Note that $\rho =\rho \left(\iota \right)$ and $\rho \left(1\right)>\rho \left(-1\right)$, as shown in Figure 6 and Figure 7.

Corollary 2.

$C_{\iota }\left(\kappa ,\eta \right)$  is the composition of a unique pair of ellipses. An ellipse is the composition only of itself with  $E_0$.

Proof. 

Let $C_{\iota }\left(\kappa ,\eta \right)=E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}$ and let $\epsilon ={\displaystyle \frac{tanh\eta }{tanh\kappa }}$. Then the theorem implies $\epsilon =tanh\left({\kappa }_2-\iota {\kappa }_1\right)$. Further, if ${tanh}^2\kappa >tanh\eta$ then $\iota =1$, equivalently, $\kappa >arctanh\epsilon$; if ${tanh}^2\kappa <tanh\eta$ then $\iota =-1$, equivalently, $\kappa <arctanh\epsilon$. Thus $\kappa ={\kappa }_2+\iota {\kappa }_1$ has the unique solution

$$\begin{array}{cc}\hfill {\kappa }_1& ={\displaystyle \frac{\iota }{2}}\left(\kappa -arctanh\epsilon \right)\hfill \\ \hfill {\kappa }_2& ={\displaystyle \frac{1}{2}}\left(\kappa +arctanh\epsilon \right)\hfill \end{array}$$

which shows, in particular, that ${\kappa }_1=0$ and ${\kappa }_2=\kappa$ if ${tanh}^2\kappa =tanh\eta$, that is, if $C_{\iota }\left(\kappa ,\eta \right)$ is an ellipse. $\square$

8. Split-inversion and Central Conics

In this section, corollaries to the Main Theorem will produce representatives for the central conics in the remaining categories listed in Section 2. We denote the split-inversion of a conic C in its focal axis by $C^{‡}$, and in its transverse axis by $C_{‡}$. We will see that the categories are related by these inversions, as summarized in Table 1. Since we reserved the terms ellipse, hyper-ellipse, and hyperbola for Steiner conics, we will add the prefix KC to the descriptions of categories C1, C2, and C4; for example, the conics in C4 are the KC-ellipses $C=C_{\iota }\left(\kappa ,\eta \right)$, each of which is the composition of a unique pair of ellipses (Corollary 2). For the C in C4, it was shown in [1] (pp. 140-142) that the $C^{‡}$ comprise C2 (KC-concave hyperbolas) and the $C_{‡}$ comprise C1 (KC-convex hyperbolas). It was also shown that each $C_{‡}$ is the locus of points ${\overline{C}}_{\iota }\left({\kappa }^{\prime },{\eta }^{\prime }\right)$ the difference of whose distances from two foci is ${\kappa }^{\prime }\ne 0$ with ${\eta }^{\prime }\ne 0$ the distance between the foci. This locus is related to $C_{\iota }\left(\kappa ,\eta \right)$ by

$$\begin{array}{cc}\hfill cosh{\kappa }^{\prime }& =coth\kappa \hfill \\ \hfill cosh{\eta }^{\prime }& =coth\eta .\hfill \end{array}$$

(16)

In particular, if $ϵ\ne 0$ then ${\left(E_{ϵ}\right)}_{‡}$ is a hyperbola, distinguished among the ${\overline{C}}_{\iota }\left({\kappa }^{\prime },{\eta }^{\prime }\right)$ by $cosh{\eta }^{\prime }={cosh}^2{\kappa }^{\prime }$, equivalently, ${\displaystyle \frac{tanh{\eta }^{\prime }}{tanh{\kappa }^{\prime }}}=\sqrt{1+{ϵ}^2}$.

Remark 4.

The KC-concave hyperbolas are also the split-inversions of the KC-convex hyperbolas in their focal axes. Specifically, ${\left({\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}_{‡}\right)}^{‡}$ is congruent to ${\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}^{‡}$ by reflection in the asymptotes of the hyperbolas ${\left(E_{ϵ}\right)}_{‡}$.

Corollary 3.

The locus ${\left(E_{ϵ}{\circ }_1{E}_{ϵ}\right)}_{‡}$  is a pair of ultra-parallel lines.

Proof.

Let ${\left(E_{ϵ}\right)}_{‡}$ represent the hyperbola, for some $0<ϵ<1$. Proposition 1 shows that that $E_{ϵ}{\circ }_1{E}_{ϵ}$ is a circle of radius $arctanhϵ$, so category C10 consists of compositions of ellipses with themselves. However, split-inversion of a circle in a diameter line is a pair of lines perpendicular to the diameter since the inversion is conformal. Thus, ${\left(E_{ϵ}{\circ }_1{E}_{ϵ}\right)}_{‡}$ is a pair of ultra-parallel lines with the focal axis of the hyperbola as the common perpendicular. $\square$

Remark 5.

With $ϵ=tanh\kappa$, the distance between the lines ${\left(E_{ϵ}{\circ }_1{E}_{ϵ}\right)}_{‡}$ is $arctanh\left(sech2\kappa \right)$. This accounts for category C9B.

The remaining categories, C9 and C9A, are obtained from the Main Theorem by continuity, as compositions of ellipses with the equidistant curves $E_1$, the boundary of $\mathcal{E}$. The proof of the following corollary is straightforward.

Corollary 4.

The composition $E_{ϵ}{\circ }_{\iota }{E}_1$ is the reducible curve represented in $\mathbb{D}$ by

$$x^2+y^2=1\pm 2x{\displaystyle \frac{1-\iota ϵ}{1+\iota ϵ}},$$

which is contained in $\mathcal{E}$ if $\iota =-1$ , and outside of $\mathcal{E}$ if $\iota =1$ . This is a pair of equidistant curves that meet the focal axis at angle $arctan{\displaystyle \frac{1+\iota ϵ}{1-\iota ϵ}}$ , from which it follows that ${\left(E_{ϵ}{\circ }_{\iota }{E}_1\right)}_{‡}$ is the pair of intersecting lines

$$y=\pm {\displaystyle \frac{1-\iota ϵ}{1+\iota ϵ}}x.$$

The constructions that produce each of the central conics are summarized in Table 1.

Table 1. Construction of Central Conics

Construction	Category	KC-Description
$E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}$	C4	ellipse
$E_{ϵ}{\circ }_{\iota }{E}_{ϵ}$	C10	circle
${\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}^{‡}$	C2	concave hyperbola
${\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}_{‡}$	C1	convex hyperbola
$E_{ϵ}{\circ }_{\iota }{E}_1$	C9	equidistant curves
${\left(E_{ϵ}{\circ }_{\iota }{E}_1\right)}_{‡}$	C9A	intersecting lines
${\left(E_{ϵ}{\circ }_1{E}_{ϵ}\right)}_{‡}:ϵ\ne 0$	C9B	ultra-parallel lines

Table 1. Construction of Central Conics

Construction	Category	KC-Description
$E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}$	C4	ellipse
$E_{ϵ}{\circ }_{\iota }{E}_{ϵ}$	C10	circle
${\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}^{‡}$	C2	concave hyperbola
${\left(E_{{ϵ}_1}{\circ }_{\iota }{E}_{{ϵ}_2}\right)}_{‡}$	C1	convex hyperbola
$E_{ϵ}{\circ }_{\iota }{E}_1$	C9	equidistant curves
${\left(E_{ϵ}{\circ }_{\iota }{E}_1\right)}_{‡}$	C9A	intersecting lines
${\left(E_{ϵ}{\circ }_1{E}_{ϵ}\right)}_{‡}:ϵ\ne 0$	C9B	ultra-parallel lines

9. The Group of Steiner Conics

The lines perpendicular to a given line L in ${\mathbb{H}}^2$ comprise an ultra-parallel pencil. Reflections in the lines of this pencil generate a subgroup $G\left(L\right)$ of collineations. Denote the identity component of this group by $G_0\left(L\right)$. The identity component consists of translations, which are products of an even number of reflections. In this section we use the Main Theorem to identify $G\left(L\right)$ with ${\mathcal{D}}_0\cup \mathcal{O}$ (see Section 1) and $G_0\left(L\right)$ with ${\mathcal{D}}_0$. In light of Remark 4, we restrict our attention to the conics with locus descriptions $C_{\iota }\left(\kappa ,\eta \right)$ and ${\overline{C}}_{\iota }\left({\kappa }^{\prime },{\eta }^{\prime }\right)$.

To obtain this representation we identify each ellipse in ${\mathcal{D}}_0$ with a translation in $G_0\left(L\right)$, and each hyperbola in $\mathcal{O}$ with a single reflection. First, with $\kappa \in \mathbb{R}$ the group of translations along L consists of the Möbius transformations

$$g\left(\kappa ,Z\right)={\displaystyle \frac{Z+itanh\kappa }{1-Zitanh\kappa }}.$$

With $ϵ=tanh\kappa$, we want $E_{ϵ}$ to represent $g\left(\kappa ,Z\right)$. Then $E_0$ would represent $g\left(0,Z\right)$ and the inverse of $E_{ϵ}$ must be defined to represent $g\left(-\kappa ,Z\right)$. We define it in accordance with $E_{ϵ}{\circ }_{-1}{E}_{ϵ}=E_0$ by giving an ellipse two orientations, $E_{ϵ}$ and $E_{-ϵ}$. With this convention, explicit reference to $\iota =\pm 1$ is no longer needed. Accordingly, $E_{{ϵ}_1}\circ E_{{ϵ}_2}$ will denote the composition C of $E_{{ϵ}_1}$ and $E_{{ϵ}_2}$, with ${ϵ}_1$ and ${ϵ}_2$ taking independent values in $\left(-1,1\right)$. Since $g\left({\kappa }_2,g\left({\kappa }_1,Z\right)\right)=g\left({\kappa }_1+{\kappa }_2,Z\right)$, we represent this translation by $E_{ϵ}$ with $ϵ={\displaystyle \frac{{ϵ}_1+{ϵ}_2}{1+{ϵ}_1{ϵ}_2}}$. Note that $E_{ϵ}$ is the unique ellipse containing the vertices of $C=E_{{ϵ}_1}\circ E_{{ϵ}_2}$ on L. We denote it by $\pi \left(C\right)$. It is implicit from the polar coordinate calculations in the proof of the Main Theorem that $\pi \left(C\right)$ is tangent to C at these points. Considering $E_{ϵ}$ to be tangent to itself, we summarize this paragraph as follows.

Proposition 3.

Let  $C=E_{{ϵ}_1}\circ E_{{ϵ}_2}$  and let  $\pi \left(C\right)$  be the unique (oriented) ellipse tangent toCat its intersections withL.Then  $\pi \left(C\right)$  represents

$g\left({\kappa }_2,g\left({\kappa }_1,Z\right)\right)$  in the commutative group  $G_0\left(L\right)$.

The elements of $G\left(L\right)$ not in the identity component are reflections. For $\kappa >0$, the involution

$$h\left(\kappa ,Z\right)={\displaystyle \frac{Z^{*}cosh\kappa +i}{cosh\kappa -Z^{*}i}}$$

is reflection in the line perpendicular to L at $i{e}^{-\kappa }$. Reflection in the line perpendicular to L at $-i{e}^{-\kappa }$ is

$$\overline{h}\left(\kappa ,Z\right)={\displaystyle \frac{Z^{*}cosh\kappa -i}{cosh\kappa +Z^{*}i}}=-h\left(\kappa ,-Z\right).$$

Since $Z\mapsto Z^{*}$ is reflection in $L^{\prime }$ we will denote it by $h\left(\infty \right)=\overline{h}\left(\infty \right)$ and refer to it as the absolute reflection. For brevity, we now express compositions in $G\left(L\right)$ in product form without Z in the notation. Thus, $g\left({\kappa }_2\right)g\left({\kappa }_1\right):=g\left({\kappa }_2,g\left({\kappa }_1,Z\right)\right)$. In particular, let ${\kappa }_j^{\prime }=arctanhsech\kappa$. Then $h\left(\kappa \right)h\left(\infty \right)=g\left({\kappa }^{\prime }\right)=h\left(\infty \right)\overline{h}\left(\kappa \right)$, and so $h\left(\infty \right)h\left(\kappa \right)=g\left(-{\kappa }^{\prime }\right)=\overline{h}\left(\kappa \right)h\left(\infty \right)$.

To extend Proposition 3 we identify each reflection with a hyperbola ${\left(E_{ϵ}\right)}_{‡}$. With $\kappa >0$ and $ϵ=tanh\kappa$, let $h\left(\kappa \right)$ and $\overline{h}\left(\kappa \right)$ be represented respectively by the oppositely oriented hyperbolas ${\left(E_{ϵ}\right)}_{‡}$ and ${\left(E_{-ϵ}\right)}_{‡}$. Equivalently, from (16), this determines a correspondence between $h\left(\kappa \right)$ and $g\left({\kappa }^{\prime }\right)$ with $tanh{\kappa }^{\prime }=sech\kappa$, and between $\overline{h}\left(\kappa \right)$ and $g\left({\kappa }^{\prime }\right)$ with $tanh{\kappa }^{\prime }=-sech\kappa$. Previously, we did not define ${\left(E_0\right)}_{‡}$ because it would consist of the absolute points $\pm i$, but here it represents $h\left(\infty \right)$ as the degenerate absolute hyperbola.

With these assignments we complete the representation of $G\left(L\right)$, beginning with products of non-absolute reflections. These are straightforward. With $tanh{\kappa }_j^{\prime }={ϵ}_j^{\prime }=sech{\kappa }_j$ in all cases, they are listed in Table 2 along with the ellipses that represent them.

Table 2. Products of Two Reflections

Reflections	Product	Representative
$h\left({\kappa }_2\right)h\left({\kappa }_1\right)$	$g\left({\kappa }_2^{\prime }-{\kappa }_1^{\prime }\right)$	$\pi \left(E_{-{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)$
$\overline{h}\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)$	$g\left({\kappa }_1^{\prime }-{\kappa }_2^{\prime }\right)$	$\pi \left(E_{{ϵ}_1^{\prime }}\circ E_{-{ϵ}_2^{\prime }}\right)$
$h\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)$	$g\left({\kappa }_1^{\prime }+{\kappa }_2^{\prime }\right)$	$\pi \left(E_{{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)$
$\overline{h}\left({\kappa }_2\right)h\left({\kappa }_1\right)$	$g\left(-{\kappa }_2^{\prime }-{\kappa }_1^{\prime },Z\right)$	$\pi \left(E_{-{ϵ}_1^{\prime }}\circ E_{-{ϵ}_2^{\prime }}\right)$

Table 2. Products of Two Reflections

Reflections	Product	Representative
$h\left({\kappa }_2\right)h\left({\kappa }_1\right)$	$g\left({\kappa }_2^{\prime }-{\kappa }_1^{\prime }\right)$	$\pi \left(E_{-{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)$
$\overline{h}\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)$	$g\left({\kappa }_1^{\prime }-{\kappa }_2^{\prime }\right)$	$\pi \left(E_{{ϵ}_1^{\prime }}\circ E_{-{ϵ}_2^{\prime }}\right)$
$h\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)$	$g\left({\kappa }_1^{\prime }+{\kappa }_2^{\prime }\right)$	$\pi \left(E_{{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)$
$\overline{h}\left({\kappa }_2\right)h\left({\kappa }_1\right)$	$g\left(-{\kappa }_2^{\prime }-{\kappa }_1^{\prime },Z\right)$	$\pi \left(E_{-{ϵ}_1^{\prime }}\circ E_{-{ϵ}_2^{\prime }}\right)$

Representing the product of a translation and a non-absolute reflection is more intricate, as indicated by the following proposition. Since split-inversion respects tangency we set $\pi \left(C_{‡}\right):=\pi {\left(C\right)}_{‡}$ for C in category C4. This is the unique (oriented) hyperbola tangent to $C_{‡}$ at its intersections with L.

Proposition 4.

If $cosh{\kappa }_{2}tanh{\kappa }_1<1$ then the reflection $h\left({\kappa }_2\right)g\left({\kappa }_1\right)$ is represented by $\pi {\left(E_{{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)}_{‡}$ with ${ϵ}_1^{\prime }=tanhln\left(e^{\kappa }+1\right)$ and ${ϵ}_2^{\prime }=-tanhln\left(e^{\kappa }-1\right)$ , where

$$\kappa =ln{\displaystyle \frac{cosh{\kappa }_{1}cosh{\kappa }_2-sinh{\kappa }_1+sinh{\kappa }_2}{cosh{\kappa }_1-cosh{\kappa }_{2}sinh{\kappa }_1}}.$$

Proof. 

First, $e^{\kappa }>1$ because ${\kappa }_2>0$. The product $h\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(t\right)$ with $cosht={\displaystyle \frac{cosh{\kappa }_2-tanh{\kappa }_1}{1-cosh{\kappa }_{2}tanh{\kappa }_1}}$. But this is equal to $cosh\kappa$, so $h\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(\kappa \right)$. We need to find ${\kappa }^{\prime }$ so that $tanh{\kappa }^{\prime }=sech\kappa$. Since $sech{\kappa }_2>tanh{\kappa }_1$, we have ${\kappa }^{\prime }=ln{\displaystyle \frac{e^{\kappa }+1}{e^{\kappa }-1}}$. Then ${\kappa }^{\prime }={\kappa }_1^{\prime }+{\kappa }_2^{\prime }$ with $tanh{\kappa }_1^{\prime }={ϵ}_1^{\prime }$ and $tanh{\kappa }_2^{\prime }={ϵ}_2^{\prime }$, whereby $\pi {\left(E_{{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)}_{‡}$ represents the product reflection. $\square$

If $cosh{\kappa }_{2}tanh{\kappa }_1>1$ then $h\left({\kappa }_2\right)g\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$ with $e^{\kappa }$ replaced by $-e^{\kappa }$. All other products are listed in Table 3. The condition that determines whether the product is $h\left(\kappa \right)$ or $\overline{h}\left(\kappa \right)$ uses the functions $p\left(j\right)=cosh{\kappa }_{1}tanh{\kappa }_2+j$ and $\tilde{p}\left(j\right)=cosh{\kappa }_{2}tanh{\kappa }_1+j$. The formula for $\kappa$ uses

$$\begin{array}{cc}\hfill {\xi }_1\left(j,k\right)& =ln\left(cosh{\kappa }_{1}cosh{\kappa }_2+jsinh{\kappa }_1+ksinh{\kappa }_2\right)\hfill \\ \hfill {\xi }_2\left(j,k\right)& =-ln\left(jcosh{\kappa }_{1}sinh{\kappa }_2+kcosh{\kappa }_2\right)\hfill \\ \hfill {\tilde{\xi }}_2\left(j,k\right)& =-ln\left(jcosh{\kappa }_{2}sinh{\kappa }_1+kcosh{\kappa }_1\right).\hfill \end{array}$$

Each product is represented by $\pi {\left(E_{{ϵ}_1^{\prime }}\circ E_{{ϵ}_2^{\prime }}\right)}_{‡}$ with ${ϵ}_1^{\prime }=tanhln\left(e^{\kappa }+1\right)$ in all cases and ${ϵ}_2^{\prime }=q\left(j\right)=tanhln{\displaystyle \frac{j}{e^{\kappa }-1}}$.

Table 3. Product of Reflection and Translation

Product	Constraint	$\kappa$	${ϵ}_2^{\prime }$
$g\left({\kappa }_2\right)h\left({\kappa }_1\right)=h\left(\kappa \right)$	$p\left(1\right)>0$	${\xi }_1\left(1,1\right)+{\xi }_2\left(1,1\right)$	$q\left(1\right)$
$g\left({\kappa }_2\right)h\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$p\left(1\right)<0$	${\xi }_1\left(1,1\right)+{\xi }_2\left(-1,-1\right)$	$q\left(1\right)$
$g\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)=h\left(\kappa \right)$	$p\left(-1\right)>0$	${\xi }_1\left(-1,-1\right)+{\xi }_2\left(1,-1\right)$	$q\left(-1\right)$
$g\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$p\left(-1\right)<0$	${\xi }_1\left(-1,-1\right)+{\xi }_2\left(-1,1\right)$	$q\left(-1\right)$
$h\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(\kappa \right)$	$\tilde{p}\left(-1\right)<0$	${\xi }_1\left(-1,1\right)+{\tilde{\xi }}_2\left(-1,1\right)$	$q\left(1\right)$
$h\left({\kappa }_2\right)g\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$\tilde{p}\left(-1\right)>0$	${\xi }_1\left(-1,1\right)+{\tilde{\xi }}_2\left(1,-1\right)$	$q\left(1\right)$
$\overline{h}\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(\kappa \right)$	$\tilde{p}\left(1\right)<0$	${\xi }_1\left(1,-1\right)+{\tilde{\xi }}_2\left(-1,-1\right)$	$q\left(-1\right)$
$\overline{h}\left({\kappa }_2\right)g\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$\tilde{p}\left(1\right)>0$	${\xi }_1\left(1,-1\right)+{\tilde{\xi }}_2\left(1,1\right)$	$q\left(-1\right)$

Table 3. Product of Reflection and Translation

Product	Constraint	$\kappa$	${ϵ}_2^{\prime }$
$g\left({\kappa }_2\right)h\left({\kappa }_1\right)=h\left(\kappa \right)$	$p\left(1\right)>0$	${\xi }_1\left(1,1\right)+{\xi }_2\left(1,1\right)$	$q\left(1\right)$
$g\left({\kappa }_2\right)h\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$p\left(1\right)<0$	${\xi }_1\left(1,1\right)+{\xi }_2\left(-1,-1\right)$	$q\left(1\right)$
$g\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)=h\left(\kappa \right)$	$p\left(-1\right)>0$	${\xi }_1\left(-1,-1\right)+{\xi }_2\left(1,-1\right)$	$q\left(-1\right)$
$g\left({\kappa }_2\right)\overline{h}\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$p\left(-1\right)<0$	${\xi }_1\left(-1,-1\right)+{\xi }_2\left(-1,1\right)$	$q\left(-1\right)$
$h\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(\kappa \right)$	$\tilde{p}\left(-1\right)<0$	${\xi }_1\left(-1,1\right)+{\tilde{\xi }}_2\left(-1,1\right)$	$q\left(1\right)$
$h\left({\kappa }_2\right)g\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$\tilde{p}\left(-1\right)>0$	${\xi }_1\left(-1,1\right)+{\tilde{\xi }}_2\left(1,-1\right)$	$q\left(1\right)$
$\overline{h}\left({\kappa }_2\right)g\left({\kappa }_1\right)=h\left(\kappa \right)$	$\tilde{p}\left(1\right)<0$	${\xi }_1\left(1,-1\right)+{\tilde{\xi }}_2\left(-1,-1\right)$	$q\left(-1\right)$
$\overline{h}\left({\kappa }_2\right)g\left({\kappa }_1\right)=\overline{h}\left(\kappa \right)$	$\tilde{p}\left(1\right)>0$	${\xi }_1\left(1,-1\right)+{\tilde{\xi }}_2\left(1,1\right)$	$q\left(-1\right)$

A fiber structure over ${\mathcal{D}}_0\cup \mathcal{O}$ will be defined in the next section. We have avoided Lie notation, such as for orthogonal groups and their indefinite forms, in our description of $G\left(L\right)$ since there will be no discussion of topology.

10. Central Conics as Fiber Elements

In this section we partition the conics in categories C4 and C1 by the Steiner conics tangent to them. Recall that every C in category C4 is the composition of a unique pair of ellipses (Corollary 2) and that the $C_{‡}$ comprise category C1.

Definition 3.

The fiber over a given ellipse $E_{ϵ}$ is the collection of C in C4 such that $\pi \left(C\right)=E_{ϵ}$. Equivalently, the set of $E_{{ϵ}_1}\circ E_{{ϵ}_2}$ with ${\kappa }_1+{\kappa }_2=\kappa$, where $ϵ=tanh\kappa$ and ${ϵ}_j=tanh{\kappa }_j$ (so the fiber over $E_0$ is just $E_0$ ).

Each element of the fiber over $E_{ϵ}$ is in the closed disk bounded by the circle $E_{{ϵ}^{\prime }}\circ E_{{ϵ}^{\prime }}$, where ${ϵ}^{\prime }=tanh{\displaystyle \frac{1}{2}}\kappa$. The fiber over $E_{{\displaystyle \frac{4}{5}}}$ is shown in Figure 8, with the ellipse in red and the circle in blue.

In the fiber over $E_{ϵ}$ let $C\left(\epsilon \right)=E_{{ϵ}_1}\circ E_{{ϵ}_2}$, where $\epsilon =tanh\left({\kappa }_2-{\kappa }_1\right)$. As with ellipses, we consider $C\left(\epsilon \right)$ and $C\left(-\epsilon \right)$ to be oppositely oriented fiber elements and define $\epsilon$ to be the (oriented) eccentricity of $C\left(\epsilon \right)$. The fiber elements are parameterized by ${\epsilon }_t=tanh\left(\kappa +t\right)$, with $t=-2{\kappa }_1$ because ${\kappa }_1+{\kappa }_2=\kappa$. Define $C\left({\epsilon }_{t_1}\right)+C\left({\epsilon }_{t_2}\right):=C\left({\epsilon }_t\right)$, where ${\epsilon }_t=tanh\left(2\kappa +t_1+t_2\right)$.Then the fiber is a group isomorphic to $G_0\left(L\right)$. The identity element is the bounding circle $C\left(0\right)$. The fiber is preserved by $G_0\left(L\right)$ with the action defined by $g·C\left({\epsilon }_t\right):=C\left({\epsilon }_t\right)+C\left(\gamma \right)$, where g is represented by the ellipse $E_{\gamma }$. Further, $G_0\left(L\right)$ acts transitively on the set of fibers, with g sending the fiber over $E_{ϵ}$ to the fiber over $\pi \left(E_{ϵ}\circ E_{\gamma }\right)$.

Proposition 5.

The conics in category C4 are partitioned into fibers over the ellipses. The group  $G_0\left(L\right)$  acts transitively on the fibers, and each fiber is a group isomorphic to  $G_0\left(L\right)$.

The conics can also be partitioned by eccentricity. Define the $\epsilon$  -section by selecting the (unique) element in each fiber with eccentricity $\epsilon$. Then $E_{\epsilon }$ is the unique ellipse in the section, as shown in Figure 9 for $\epsilon ={\displaystyle \frac{1}{2}}$.

For $ϵ\ne 0$, the fiber over a hyperbola ${\left(E_{ϵ}\right)}_{‡}$ is the split-inversion of the fiber over $E_{ϵ}$. Figure 10 shows the fiber over ${\left(E_{{\displaystyle \frac{4}{5}}}\right)}_{‡}$, with the hyperbola in red. The elements are bounded by ${\left(E_{{\displaystyle \frac{1}{2}}}\circ E_{{\displaystyle \frac{1}{2}}}\right)}_{‡}$, which is in category C9B by Corollary 3. In this example, the distance between these ultra-parallel lines (blue) is $ln2$ (see Remark 5).

The KC-convex hyperbolas in the fiber over ${\left(E_{ϵ}\right)}_{‡}$ can be assigned eccentricities greater than 1 in accordance with (16) and consistent with the fiber over $E_{ϵ}$. A section for a given eccentricity would then be the split inversion of an $\epsilon$ -section of KC-ellipses.

11. Conclusions

In Jakob Steiner’s preface to [2] he states

Here the main thing is neither the synthetic nor the analytic method, but the discovery of the mutual dependence of the figures and of the way in which their properties are carried over from the simpler to the more complex ones.

The source of this quote has been the inspiration for the constructions in this article. These constructions were obtained intrinsically, within conformal models of the hyperbolic plane, from the incidence properties of ${\mathbb{H}}^2$ and the action of its collineation group. This is in the spirit of Steiner’s construction of conics. We used elliptic curve addition on the orthogonal trajectories of the central Steiner conics to generate all central conics, which in turn imposed a group structure on the Steiner conics themselves.

The underlying symmetries of the central conics made these constructions possible. Though we previously classified all Steiner conics in ${\mathbb{H}}^2$ we are not aware of analogous results for the categories of non-central conics. Many of these are of so-called parabolic type, but there does not appear to be an inversive representation of the Steiner parabolas for which their orthogonal trajectories have a simple addition structure. Perhaps this work will encourage investigation in that direction.