Quadratic Hyperboloids in Minkowski Geometries

A Minkowski plane is Euclidean if and only if at least one hyperbola is a quadric. We discuss the higher dimensional consequences too.


Introduction
Let I be an open, strictly convex, bounded domain in R n , (centrally) symmetric to the origin. Then function d : R n × R n → R defined by where a < d(F 1 , F 2 )/2, is called a hyperbola if n = 2, and a hyperboloid in higher dimensions, where F 1 , F 2 ∈ R n are called the focuses, and a > 0 is called the radius.
A hypersurface in R n is called a quadric if it is the zero set of an irreducible polynomial of degree two in n variables. We call a hypersurface quadratic if it is part of a quadric. Since every isometric mapping between two Minkowski geometries is a restriction of an affinity, and every affinity maps quadrics to quadrics, the quadraticity of a metrically defined hypersurface is a geometric property in each Minkowski geometry. Thus, the question arises

Notations and Preliminaries
Points of R n are labeled as A, B, . . . , vectors are denoted by − − → AB or a, b, . . . , but we use these latter notations also for points if the origin is fixed. The open segment with end points A and B is denoted by AB, while AB denotes the open ray starting from A passing through B, finally, AB = AB ∪ AB.
On an affine plane, the affine ratio (A, B; C) of the collinear points It is easy to observe in D1 that a hyperboloid intersects line F 1 F 2 , the main axis, in exactly two points, whose distance is twice the radius. Further notions are the (linear) eccentricity c = d(F 1 , F 2 )/2, the numerical eccentricity ε = c/a. The metric midpoint of the segment F 1 F 2 is called the center.
Notations u ϕ = (cos ϕ, sin ϕ) and u ⊥ ϕ := (cos(ϕ + π/2), sin(ϕ + π/2)) are frequently used. It is worth noting that, by these, we have d dϕ u ϕ = u ⊥ ϕ . A quadric in the plane has the equation of the form Q σ s := (x, y) : in a suitable affine coordinate system s, and we call it elliptic, parabolic, or hyperbolic, if σ = 1, σ = 0, or σ = −1, respectively. We usually polar parameterize the boundary ∂D of a non-empty domain D in R 2 starlike with respect to a point P ∈ D so that r : [−π, π) → R 2 is defined by r(ϕ) = r(ϕ)u ϕ , where r is the radial function of D with base point P .
We call a curve analytic if the coordinates of its points depend on its arc length analytically.

Quadratic Hyperboloids in Minkowski Geometries
hence the function s : ξ → s(ξ) is strictly monotonously increasing, and therefore its inverse function σ : s(ξ) → ξ exists and is strictly monotonously increasing. First, assume the analyticity of r. Then, as r is bounded from below by a positive number, the integrand on the right-hand side of (3.1) is analytic, and therefore s is analytic. Asṡ(ξ) is positive by (3.1), the analyticity of σ follows from the analytic inverse function theorem [3,Theorem 4.2], and this implies the analyticity of p(s) = r(σ(s)) = r(σ(s))u σ(s) .
The lemma is proved.
Notice that the differentiation of the last formula in the proof and then the substitution of the derivative of (3.1) givė Let H be a hyperbola with center O and focuses F 1 and F 2 . Let us label the intersection points of F 1 F 2 and H so that There exists an angle Φ ∈ (0, π/2) such that a unique point H exists on Assuming that H 2i is defined for an i ∈ N, we define sequences recursively as follows (see Fig. 1 Lemma 3.2. If i → ∞, then α 2i and ϕ 2i tend to zero, β 2i , β 2i+1 , and ϕ 2i+1 tend to π, and α 2i+2 /α 2i tends to (F 1 , F 2 ; A, B).
Let r 1 and r 2 be curves in the plane with analytic arc length parametrization on [−1, 1] such that r 1 (0) = r 2 (0) andṙ 1 (0) =ṙ 2 (0). Let be the line through r 1 (0) that is orthogonal toṙ 1 (0), and let F 1 , F 2 , and B be different points on such that B ∈ F 1 F 2 and r 1 (0) / ∈ {B, F 1 , F 2 }. Let h be an analytic arc length parameterization of a curve such that B = h(0) and  Letting H ⊥ be the orthogonal projection of H onto , L'Hôpital's rule gives δ(γ) exists, then L'Hôpital's rule can be used, so we get This proves the lemma.

One Hyperbola in a Minkowski Plane
We start by considering the Minkowski plane (R 2 , d I ) with indicatrix I.  A, B and I, J, respectively. See Fig. 3. Given is a quadric, ϕ and H are bijectively related, hence the functions α(ϕ), β(ϕ) are well defined.
The symmetry of I entails that t I t J , and it also follows that the affine center of the quadric H a dI ;F1,F2 coincides with its metric center O, hence t A t B too.

Lemma 4.2.
If the hyperbola H a dI ;F1,F2 is an analytic curve in a neighborhood of A and B, then the curve ∂I O is analytic in a neighborhood of I and J.
Proof. By Lemma 3.1 and its proof, the functions h 1 , h 2 , the angles α(s), β(s), and the inverses of the angles, where s is the arc length parameter, are clearly analytic, hence we deduce that β(α) and α(β) are also analytic functions.
As x → 1/x is analytic in a neighborhood of 1, to prove that r(α) is analytic in a neighborhood of 0, it is enough to prove thatr(α) := 1/r(α) is analytic in some neighborhood of 0. Bearing this in mind, we reformulate (4.1) asr .
. However, by [3,Theorem 4.6], such a functional equation has a unique solution, which additionally is analytic in a neighborhood of 0. Consequently, r(α) is the reciprocal of that unique analytic solution, so ∂I O is analytic around J, and, by its symmetry, around I too.

Theorem 4.3. A Minkowski plane is a model of the Euclidean plane if and only if at least one hyperbola is a quadric.
Proof. As every hyperbola is a quadric in the Euclidean plane, we only have to prove that a Minkowski plane is Euclidean if at least one hyperbola is a quadric.
Assume that H a dI ;F1,F2 is a quadric.
Being a hyperbolic quadric, H a dI ;F1,F2 has two asymptotes + and − through O. Let C 1 and C 2 be the points where they intersect the straight line t A . Let C denote the unit circle of d e . See Fig. 4. Then both H a de;F1,F2 and H a dI ;F1,F2 are hyperbolic quadrics, and have two common tangents t A and t B , two common asymptotes, and two common points A and B, hence they coincide.
We use the notations introduced in the previous sections, and consider the hyperbola H a dI ;F1,F2 . Fix an arbitrary point H 0 ∈ H a dI ;F1,F2 , and let the point R i ∈ I (i = 1, 2) be such that OR i F i H 0 . Let the straight line t i (i = 1, 2)be tangent to I