An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

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Abstract. In this paper, we introduce a compactification of the moduli space of polynomial maps with a fixed degree n (≥ 2) such that the map from it to P n−1 (C) defined by using the elementary symmetric functions of multipliers at fixed points is a continuous surjection. Introduction and main theoremsLet n ≥ 2 and Poly n be the set of all polynomial maps of C to itself with degree n. We say that two maps p 1 and p 2 in Poly n are affine conjugate if there exists a biholomorphic automorphism g of C such that gThe moduli space of polynomial maps with degree n is the set of all affine conjugacy classes of maps in Poly n , and is denoted by MPoly n . Here, recall that a natural complex orbifold structure can be introduced on MPoly n . Remark 1.1. These spaces were investigated by Branner and Hubbard [2], [3], and Milnor [12] in the case of degree 3, and then by the first author in general cases (cf.[9]).Several kinds of compactification of MPoly n have been considered. One is given as the closure in the GIT compactification of the moduli space of rational maps, which is defined in [14]. (See also [4].) DeMarco and McMullen introduced one by using tree representations ([5]), and also DeMarco discussed another one in [4]. Here, we show that the extended moduli space introduced by the first author in [8] (also cf. [9], Introduction) gives a natural compactification of MPoly n with a suitable topology. By identifying the affine conjugacy class p of p ∈ Poly n with the collection { p } consisting of this class only, we consider MPoly n as a subset of M n .Here, by definition, an element X of M n is an unordered set of affine conjugacy classes. But for the sake of convenience, we fix an order temporarily, and represent . Then, we can associate X with a collection of sets of numbers as follows. For X = { p j } J j=1 , assume that the fixed points of p j , counted including multiplicities, are represented by numbersfor every j. We call the partition of {1, · · · , n} into

Answering a question about triangle inequality suggested by R. Li, A. Barrlund [3] introduced a distance function which is a metric on a subdomain of R n . We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.