LIPSCHITZ CONSTANTS AND QUADRUPLE SYMMETRIZATION BY M ¨OBIUS TRANSFORMATIONS

. Due to the invariance properties of cross-ratio, M¨obius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a M¨obius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the M¨obius transformation in the Euclidean or the chordal metric.


Introduction
Symmetry is an important property considered by many mathematicans working in various areas of research, including classical analysis, geometry, and solutions to extremal problems [3,4,6,7].Problem-specific methods have been introduced with the purpose of transforming objects into symmetric ones [3].Another use of symmetry is to reduce some complicated problem to its special case which exhibits symmetry and is therefore easier to solve than the original question.
For quadruples of points on the complex plane, a symmetrization can be performed by means of a Möbius transformation.The use of Möbius transformations is in fact a standard tool in classical function theory.However, while there are several metrics, such as the hyperbolic metric, that are invariant with respect to the Möbius transformations, these mappings affect the Euclidean distances between the complex points considered.Our aim is to analyze "cost" of the symmetrization due to distortion under Möbius transformations, measured in terms of Lipschitz constants.These Lipschitz constants can then be formulated with respect to Euclidean, hyperbolic, or chordal geometry.
We consider symmetrization of quadruples of points in several cases, under various constraints of the location of the points of the quadruple {a, b, c, d}.In the most general case, we find a Möbius transformation h mapping the quadruple onto the quadruple {−1, −y, y, 1} symmetric with respect to the origin.For any such Möbius transformation h, it seems natural to call h −1 (0) a Möbius center of the quadruple.We give an explicit formula for such a Möbius center.
where y is as in Lemma 4.2.
The structure of this paper is as follows.In Section 3, we study a certain chordal isometry and related Lipschitz constants of the chordal metric.In Section 4, we find a Möbius transformation for the symmetrization of any four distinct points on the complex plane and prove Theorem 1.1.In Section 5, we introduce three normalization methods for a pair of points in the unit disk and, in Section 6, we consider the symmetrization of quadruples of points on the unit circle.

Preliminaries
Let us first introduce the notations used in this paper.For a dimension n ≥ 0, e n is the nth unit vector of the real space R n , B n is the unit ball, and S n−1 is the unit sphere.For a point x ∈ R n and a radius r, let B n (x, r) = {y ∈ R n : |x − y| < r} and S n−1 (x, r) = {y ∈ R n : |x − y| = r}.For any point x ∈ C, denote its complex conjugate by x.For two distinct points x, y ∈ R n , let L(x, y) be the Euclidean line passing through them.Denote the extended real space by R n = R n ∪ {∞}.
and a 1 , a, b, b 1 are ordered in such a way that |a 1 − a| < |a 1 − b|.
where • denotes the dot product.The reflection in this hyperplane is defined by the function h : R

Define then the inversion in
A Möbius transformation is a function f : R n → R n that can be defined as a function with an integer m ≥ 1 so that each h j , j = 1, ..., m, is either a reflection in some hyperplane or an inversion in a sphere.The Möbius transformation is called sense-preserving when m here is even and sense-reversing when m is odd.By Liouville's theorem, any conformal mapping with a domain in R n , n ≥ 3, is a Möbius transformation in R n restricted to this domain [11,Rmk 3.44,p. 47].
In the two-dimensional case, the expression of a sense-preserving Möbius transformation can be written as The expression of the sense-reversing Möbius transformation can be obtained by replacing z by z in (2.6).The special Möbius transformation Ahlfors [1] gave the following factorization for T a , a ∈ B 2 \ {0}, as a composition of an inversion followed by a reflection which is given here in the complex notation.Denote a * = a/|a| 2 .Let r = |a| −2 − 1 and σ a be the inversion in the circle S 1 (a * , r) orthogonal to the unit circle S 1 .Then σ a (a) = 0 and σ a (a * ) = ∞.Let p a be the reflection over the line through 0 and perpendicular to a. Then both σ a and p a are sense-reversing transformations and their composite transformation T a in (2.7) is a sense-preserving Möbius transformation with the decomposition Clearly, p a is a Euclidean isometry and hence the information about distance distortion under T a is due to the mapping σ a .Moreover, for a, b The above simple formula for the point c follows from Proposition 2. Theorem 2.9.Let T a and T a,b be the above Möbius transformations (2.8).Then where the Lipschitz constant are defined for the Euclidean metric.
2.12.Hyperbolic geometry.Denote hyperbolic sine, cosine and tangent by sh, ch, and th, respectively.The hyperbolic metric ρ B n in the unit ball B n is defined by , We denote the hyperbolic line through .

Lipschitz constants in chordal metric
In this section, we introduce an isometry t m of the chordal metric.Note that, while the mapping T a is an isometry for the hyperbolic metric, it is not a chordal isometry.However, our computer tests suggest that it has the following Lipschitz constant.Conjecture 3.1.For a ∈ B n , the Lipschitz constant of the mapping T a in the chordal metric is It is well-known that (1 + |a|)/(1 − |a|) is also the Lipschitz constant Lip(T a |B n ) for the Euclidean metric, as stated in Theorem 2.9.
Let us then define the chordal isometry.Recall the notation a * for a ∈ R 2 \ {0} and p a from Subsection 2.5.Let s a be the inversion in the circle S 1 (−a * , 1 + |a| −2 ).Now, we can define a chordal isometry as [11, (3.23), p. 40] Trivially, t a is a Möbius transformation and it satisfies t a (a) = 0.In fact, we have the following formula [11, (B.9) We see from the above formula that, if a, b ∈ B 2 , their chordal midpoint m ∈ B 2 .
Corollary 3.4.For a, b ∈ R 2 , let m be their chordal midpoint as in Theorem 3.3.Then the chordal isometry t m : R 2 → R 2 has the formula Furthermore, we have Proof.The result follows by inputting m as in Theorem 3.3 into t m (z) = (z − m)/(1 + mz).

Möbius symmetrization
In this section, we prove a few results related to using a Möbius transformation for the symmetrization of quadruples of points on the complex plane. Otherwise, where The two possible solutions for y above are reciprocal numbers so we can choose y so that |y| < 1 in this case.
Proof.First, if a, b, c, d are collinear and satisfy |a − b| = |c − d|, then h is a linear map and can be written Thus, the first part of the lemma follows.
Next, let us consider the other case.We may assume that h(z) = pz + q z + s since h is clearly not a linear map.From the conditions h(a By eliminating q from (4.7) and (4.9), we have Similarly, by eliminating p from (4.7) and (4.9), we have By eliminating s from (4.8) and (4.10), we have Elimination of p from (4.8) and (4.10) gives us Finally, by eliminating p from Y 1 = 0 and Y 2 = 0, we have which is equivalent to k 0 y 2 + 2k 1 + k 0 = 0 where k 0 and k 1 are as in the lemma.The latter part of our lemma follows from this.
The Möbius transformation of the former lemma is a very useful mapping as it can be used for symmetrization.Namely, with this transformation, we can map the arbitrary quadruple of points a, b, c, d onto the points −1, y, −y, 1 that are clearly symmetric with respect to the origin.However, as the origin is not preserved under the mapping h, this raises the question about the location of the Möbius center h −1 (0) in terms of a, b, c, d.We can now prove Theorem 1.1 from Introduction.

Proof of Theorem 1.1
Proof.The mapping h is as in Lemma 4.2.If a, b, c, d are collinear and |a − b| = |c − d|, then it is a linear map and h −1 (0) is the midpoint of a and d.Consider then the other case.Denote w = h −1 (0).Now, for p, q, s as in Lemma 4.2,

Three normalizations of point pairs
Mathematical proofs involving distances are often simplified if the configurations studied exhibit some kind of symmetry or special structure.We study here, for a given pair of distinct points a, b in B 2 \ {0} , three natural normalizations: (1) , b, 0 are collinear, (3) |a| = |b| .If the points are in general position, then they can be mapped onto each of the above special positions by means of a Möbius transformation.The cost is that the geometry is changed in each case: we measure the cost in terms of the Lipschitz constant of the Möbius transformation.
A simple way to achieve the normalization (2) above is to relabel the points a and b such that |a| ≤ |b| and to use the mapping T a .By Theorem 2.9 the cost of normalization can be immediately obtained.In our discussion below, the plan is to carry out the normalization with a smaller cost, if possible.
In other words, Lip(T h ab ,k ) is sometimes less than the other two Lipschitz constants and sometimes greater than them.

Quadruples of points on the unit circle
Consider four distinct points a, b, c, d on the unit circle in the positive order.We can symmetrize the quadruple in two different ways using a Möbius transformation of the unit disk: (1) After the symmetrization, the new quadruple is symmetric with respect to a diameter of the unit disk.(2) After the symmetrization, the points of the new quadruple form the vertices of a rectangle.We will next show how to construct the corresponding Möbius transformations and estimate the cost of symmetrization, the Lipschitz constant of the transformation.Theorem 6.1.[13] (1) Let Then the point w 2 is the orthocenter of the triangle with vertices at the points 0, w 1 , w 3 .
( For a given point w 1 ∈ R 2 \ B 2 , a circle S 1 (r, w 1 ), r > 0, is orthogonal to the unit circle if and only if its radius r is |w 1 | 2 − 1.Consequently, the point w 5 in Theorem 6.2 has the formula  they are spread more evenly around the whole unit circle.This observation could be also concluded from the geometric meaning of the points w 4 and w 5 .

Theorem 1 . 1 .
Let h be the Möbius transformation of R 2 that maps distinct points a, b, c, d ∈ R 2 onto −1, y, −y, 1, respectively.If a, b, c, d are collinear and |a − b| = |c − d|, then h −1 (0) = (a + d)/2.Otherwise, The intersection point of two non-parallel lines L(a, b) and L(c, d) is given by [11, Ex. 4.3(1), p. 57 & p. 373] (2.1) LIS[a, b, c, d] = (ab − ab)(c − d) − (cd − cd)(a − b) (a − b)(c − d) − (c − d)(a − b) .If the four points a, b, c, d are on the unit circle, then we have z = 1/z for z ∈ {a, b, c, d} and the formula (2.1) can be simplified as in the following proposition.Proposition 2.2.[12, p.13]If a, b, c, d are four distinct complex points on the unit circle S 1 chosen so that the lines L(a, b) and L(c, d) are non-parallel, then the complex conjugate f of the intersection point f of the lines L[a, c] and L[b, d]

3 ( 2 )
for the intersection of two lines c = LIS[a, b, a * , b * ].For a, b ∈ B 2 \ {0}, the Möbius transformation T a,b is a reflection over the line L(0, b) composed with the inversion in S 1 (c, r) where r = |c| 2 − 1 so that S 1 (c, r) is orthogonal to the unit circle.See Figure 1.

Figure 1 .
Figure 1.The points a, b, a * , b * when a = −0.7iand b = 0.5, the intersection point c = LIS[a, b, a * , b * ] of the lines L(a, b) and L(a * , b * ), the unit circle, the circle S 1 (c, r) for r = |c| 2 − 1, and the hyperbolic line J * [a, b].The end points of J * [a, b] are collinear with c, as denoted with dashed line.The Möbius transformation T a,b is a reflection over the line L(0, b) composed with the inversion in the circle S 1 (c, r).

Theorem 5 . 1 .
Consider a given pair of distinct points a, b in B 2 \ {0}.Let k be the point on the hyperbolic line J * [a, b] that is closest to the origin.In other words, k = 0 if a, b, 0 are collinear and, if a, b, 0 are not collinear, then k = o ab t (t − √ t 2 − 1), t = |o ab |, where let o ab be the center of the circle through a and b, orthogonal to the unit circle.Let h ab be the hyperbolic midpoint of a and b.Then there are Möbius transformations m j : B 2 → B 2 , j = 1, 2, 3 with the following properties: m 1 (a) = −m 1 (b) with (5.2) Lip(m 1 ) ≤ Lip(T h ab ) = 1 + |h ab | 1 − |h ab | , m 2 (a), m 2 (b), 0 are collinear and (5.3) Lip(m 2 ) ≤ Lip(T k ) = 1 + |k| 1 − |k| , and |m 3 (a)| = |m 3 (b)| and(5.4)Lip(m 3 ) ≤ Lip(T h ab ,k ),where T h ab ,k is the mapping defined in (2.8).Proof.We can choose for instance the mappings m 1 = T h ab , m 2 = T k , and m 3 = T h ab ,k in which case the following inequalities of the Lipschitz constants Lip(m j ) presented in the theorem hold with equality.For all a, b ∈ B 2 \ {0}, |k| ≤ |h ab | where h ab and k are as in Theorem 5.1.Namely, the hyperbolic point h ab trivially is on the hyperbolic line J * [a, b], but, unlike k, h ab is 0

Figure 2 .Figure 3 .
Figure 2. The Möbius transformation T k maps the nearest point k on the hyperbolic line J * [a, b] through a = −0.9iand b = 0.5 − 0.3i to the origin so that the images of a and b are collinear with the origin.

2 ) 1 . 6 . 2 .
The point of intersection of the hyperbolic lines J * [a, c] and J * [b, d] is given byw 4 = (ac − bd) ± (a − b)(b − c)(c − d)(d − a) a − b + c − d ,where the sign "+" or "−" in front of the square root is chosen such that |w 4 | < Theorem Let a, b, c, d be four distinct points on the unit circle in the positive order.Let w 4 be the point in Theorem 6.1(2) and w 5 be the midpoint of the orthogonal arc with center at w 1 = LIS[a, b, c, d].There exists a Möbius transformation m 4 : B 2 → B 2 such that m 4 (a), m 4 (b), m 4 (c), m 4 (d) are symmetric with respect to a line through 0 and (6.3) Lip(m 4 ) = Lip(T w 4 ,w 5 ).Moreover, if m 5 = T w 4 , then m 5 (a) = −m 5 (c), m 5 (b) = −m 5 (d).

4 )
The intersection point w 1 = LIS[a, b, c, d] can be found with Proposition 2.2.If a = e 1i , b = e 3i , c = e 4i , and d = e 5i , we have Lip(T w 4 ,w 5 ) ≈ 1.496, Lip(T w 4 ) ≈ 1.833, and, if a = e 6i , b = e 6.1i , c = e 6.13i , and d = e 6.15i , we have Lip(T w 4 ,w 5 ) ≈ 449.5, Lip(T w 4 ) ≈ 60.00.Consequently, neither of the mappings in Theorem 6.2 is always better than the other in terms of its Lipschitz constant.Computational tests suggest that T w 4 produces a smaller Lipschitz constant if the points a, b, c, d are close to each other, while T w 4 ,w 5 is better when

Figure 4 .
Figure 4.The points w 4 and w 5 defining the Möbius transformation T w 4 ,w 5 of Theorem 6.2 when a = 1, b = e 0.3i , c = e 1.5i , and d = e 2.1i .