**Horocyclic Coordinates:**

$$\mathbf{x}\_0 = \cosh w + \frac{1}{2} \zeta^2 e^w,\tag{57a}$$

$$\mathbf{x}\_1 = \boldsymbol{\zeta}^\* \boldsymbol{e}^w,\tag{57b} \\ \mathbf{x} = \mathbf{x}\_1 \mathbf{y} \in \mathbb{R} \\ \text{and} \\ \mathbf{y} = \mathbf{y}\_1 \mathbf{x}\_1 \mathbf{y} \tag{57b}$$

$$\mathbf{x}\_2 = \sinh w - \frac{1}{2} \zeta^2 \ e^w. \tag{57c}$$

Even in this case, it is straightforward to verify that *x*20 − *x*21 − *x*22 = 1, and *x*0 1; accordingly, the horocyclic coordinates (57) are appropriate for describing the upper sheet <sup>2</sup>*X*+2 of the real two-sheeted hyperboloid. In particular, we focus our attention on the meridian section <sup>2</sup>*X*-<sup>+</sup> of the upper sheet <sup>2</sup>*X*+2 , which lies in the plane *x*1 = 0 and whose equation is *x*20 − *x*22 = 1. We consider the intersections of <sup>2</sup>*X*-<sup>+</sup> with the family of planes <sup>2</sup>*Pw* with equation *x*0 + *x*2 = *e<sup>w</sup>* (*w* ∈ R), i.e., the points <sup>2</sup>*xw*, whose coordinates are *x*0 = cosh *w*, *x*1 = 0, *x*2 = sinh *w*. The sections of <sup>2</sup>*X*+2 by these planes are the (real) parabolae <sup>2</sup>Π+*w* . Setting *ζ* = 0 in (57), we obtain the point <sup>2</sup>*xw* (*w* ∈ R), which is the apex of the corresponding parabola <sup>2</sup>Π+*w* (see Figure 4).

**Remark 5.** *As we shall see in the next section, the parabolae* <sup>2</sup>Π+*w generate (through a stereographic projection on the non-Euclidean disk) a family of horocycles, represented by the Euclidean circles (illustrated in Figure 4) which are tangent to the boundary of the unit disk at the point x*1 = 0, *x*2 = <sup>−</sup>1*. It is worth emphasizing that this is the sole family of horocycles which is generated by the fibration illustrated above. On the other hand, the non-Euclidean Fourier transform, which will be studied in the next section, demands considering in the non-Euclidean disk all the families of horocycles which are obtained by moving the point of tangency along the entire horizon from 0 to* 2*π. We shall see in Lemma 2 of Section 3.4.1 how this difficulty will be overcome (see also Remark 6).*

### *3.3. Stereographic Projection from the Upper Sheet of the Two-Sheeted Hyperboloid to the Non-Euclidean Unit Disk*

Let *u* = (*<sup>u</sup>*1, *<sup>u</sup>*2) denote the Cartesian coordinate system in the open unit disk *D* .= {*u* : *u*11 + *u*22 < <sup>1</sup>}, which lies in the plane *x*0 = 0:

$$u\_1 = \tanh\left(\frac{v}{2}\right)\sin\psi \qquad\qquad (v \gg 0, \psi \in [0, 2\pi)),\tag{58a}$$

$$
\mu\_2 = \tanh\left(\frac{\upsilon}{2}\right) \cos \psi. \tag{58b}
$$

Then: |*u*|<sup>2</sup> = *u*21 + *u*22 = tanh<sup>2</sup>(*v*/2). Let *B* .= {*u* : *u*11 + *u*22 = 1} denote the boundary of *D*, i.e., the horizon. Consider the Riemannian structure

$$\mathrm{d}s^2 = \frac{4(\mathrm{d}u\_1^2 + \mathrm{d}u\_2^2)}{[1 - (u\_1^2 + u\_2^2)]^2} = \mathrm{d}v^2 + \sinh^2 v \,\mathrm{d}\psi^2,\tag{59}$$

where d*v*<sup>2</sup> + sinh2*v* d*ψ*<sup>2</sup> can also be obtained from (56) through the equality: d*s*<sup>2</sup> = −d*x*20 + <sup>d</sup>*x*21 + <sup>d</sup>*x*22. Let us embed the non-Euclidean disk in the complex *z*-plane so that each point of the unit disk can be represented either by the coordinates (*<sup>u</sup>*1, *<sup>u</sup>*2) in (58) or through the polar representation of *z*, which in the present case reads: *z* = |*z*| exp[i( *π*2 − *ψ*)], |*z*| = tanh( *v*2 ). Note that in this latter representation we are forced to write exp[i( *π*2 − *ψ*)] with *ψ* induced by (56) (instead of the standard expression exp(<sup>i</sup>*ψ*)) in order to have the correct correspondence between Cartesian and polar representations. Accordingly, the points *b* of the boundary *B* of the non-Euclidean disk will be described by *b* = exp[i( *π*2 − *φ*)], *φ* being defined analogously to *ψ* that is, measured from the positive *<sup>u</sup>*2-axis and increasing toward the positive *<sup>u</sup>*1-axis. The Riemannian structure (59) induces the usual non-Euclidean distance on *D*:

$$d(0, z) = \ln \frac{1 + |z|}{1 - |z|} \qquad \qquad z = |z| \, \epsilon^{\mathbf{i}(\frac{\pi}{2} - \varphi)} \,, $$

where |*z*| = tanh(*v*/2) and 0 is the center of the unit disk. In *D*, the geodesics are circular arcs intersecting the unit circle at right angles. In particular, all diameters of the unit circle are straight lines since these diameters can be considered as arcs of infinite large radius. A pencil of parallel straight lines is given by arcs of Euclidean circles orthogonal to the unit circle, lying in its interior and intersecting the boundary *B* at a common point *b*. The lines orthogonal to this pencil of parallel geodesics are the circles tangent from within to the horizon at the point *b*. Since these circles are the Euclidean images of the horocycles, we shall refer to the point of contact *b* as the normal to the horocycle.

The coordinates (*<sup>x</sup>*0, *x*1, *<sup>x</sup>*2) that describe the point *P* varying on the upper sheet <sup>2</sup>*X*+2 of the two-sheeted hyperboloid are related to the coordinates (*<sup>u</sup>*1, *<sup>u</sup>*2) in the open non-Euclidean unit disk *D* through the following formulas (Lemma 1, p. 49, [25]):

$$\mathbf{x}\_0 = \frac{1+|u|^2}{1-|u|^2}, \qquad \mathbf{x}\_1 = \frac{2u\_1}{1-|u|^2}, \qquad \mathbf{x}\_2 = \frac{2u\_2}{1-|u|^2}.\tag{60}$$

Each line of intersection of <sup>2</sup>*X*+2 with the plane *ax*0 + *bx*1 + *cx*2 = 0 (*<sup>a</sup>*, *b*, *c* ∈ R) is mapped by the transformation (60) into a circular arc intersecting the horizon at right angles (Lemma 3, p. 51, [25]). In order to find the curve into which a straight line on <sup>2</sup>*X*+2 is mapped, it is sufficient to substitute into

the equation of the plane *ax*0 + *bx*1 + *cx*2 = 0 the explicit expression (60) of the variables *xi* (*i* = 0, 1, 2) in terms of *u*1, *u*2. Then, the equation

$$a\frac{1+|u|^2}{1-|u|^2} + \frac{2bu\_1}{1-|u|^2} + \frac{2cu\_2}{1-|u|^2} = 0\tag{61}$$

is reduced after simple algebra to the equation of the circle

$$\left(u\_1 + \frac{b}{a}\right)^2 + \left(u\_2 + \frac{c}{a}\right)^2 = \frac{b^2 + c^2 - a^2}{a^2} \qquad (a \neq 0),\tag{62}$$

with radius *r* = √*b*<sup>2</sup> + *c*2 − *a*2/|*a*| and center at the point (−*b*/*<sup>a</sup>*, −*<sup>c</sup>*/*a*) if *a* = 0, and to the line *bu*1 + *cu*2 = 0 if *a* = 0. Note that the image into *D* of a straight line on <sup>2</sup>*X*+2 is not the entire circle (62) but only its part contained in the unit disk.

Let us now map the hyperbola <sup>2</sup>*X*-<sup>+</sup> = <sup>2</sup>*X*+2 ∩ {*<sup>x</sup>*1 = <sup>0</sup>}, namely, the meridian of the upper sheet of the two-sheeted hyperboloid, into the non-Euclidean unit disk. By setting *a* = *c* = 0 and *b* = 1 in the equation of the planes given above, from (61), we see that <sup>2</sup>*X*-<sup>+</sup> is mapped into the diameter *u*1 = 0 of the non-Euclidean disk: precisely (see (58)), we have the diameter *u*1 = 0, *u*2 = tanh(*v*/2), which tends to *u*2 = +1 (i.e., *z* = +i) for *v* → +∞ and *ψ* = 0, and to *u*2 = −1 (i.e., *z* = −i) for *v* → +∞ and *ψ* = *π* (see Figure 4).

Next, we map the parabola <sup>2</sup>Π+*w* , with apex <sup>2</sup>*xw* = (cosh *w*, 0, sinh *w*) lying on the meridian <sup>2</sup>*X*-+, which is represented in horocyclic coordinates by Equation (57). The apex <sup>2</sup>*xw* of the parabola lies on the right branch *x*2 > 0 of the meridian <sup>2</sup>*X*-<sup>+</sup> for *w* > 0, and on the left branch *x*2 < 0 for *w* < 0; for *w* = 0, the apex of the parabola has coordinates (1, 0, <sup>0</sup>). These parabolae are generated by the intersection of <sup>2</sup>*X*+2 with the plane <sup>2</sup>*Pw* whose equation is: *x*0 + *x*2 = *<sup>e</sup>w*. By substituting in this latter equation the expressions of *x*0 and *x*2 given in (60), we obtain:

$$u\_1^2 + \left(u\_2 + \frac{1}{1 + \epsilon^{w}}\right)^2 = \frac{\epsilon^{2w}}{(1 + \epsilon^w)^2},\tag{63}$$

which represents a circle. In view of (63), we can thus say that the mapping of the parabolae <sup>2</sup>Π+*w* into the unit non-Euclidean disk are the Euclidean circles (images of horocycles) with center in (*<sup>u</sup>*1 = 0, *u*2 = −(<sup>1</sup> + *<sup>e</sup><sup>w</sup>*)−<sup>1</sup>), radius *rw* = (1 + *e*<sup>−</sup>*<sup>w</sup>*)−1, and tangent from within to the horizon at the point (*<sup>u</sup>*1 = 0, *u*2 = −<sup>1</sup>) (i.e., *z* = −i; see Figure 4). Moreover, these circles cut orthogonally the diameter *u*1 = 0 of *D* in the point (*<sup>u</sup>*1 = 0, *u*2 = tanh(*w*/2)), which lies above the center of *D* when *w* > 0 and below the center of *D* when *w* < 0.

### *3.4. Connection between Spherical Laplace Transform and Non-Euclidean Fourier Transform*
