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Masayo Fujimura

Publications and source records attributed to Masayo Fujimura.

15 recordsLinked to original sources
arXiv2025

Hilbert metric and H\"older continuity of quasiregular mappings

We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk $\mathbb{B}^2$ onto a bounded convex domain $D$. The main result deals with the H\"older continuity of these mappings with respect to Hilbert metrics of $\mathbb{B}^2$ and $D$. Also several open problems are formulated.

math.CV↗
arXiv2025

Hilbert metric and quasiconformal mappings

We prove a functional identity between the Hilbert metric and the visual angle metric in the unit disk. The proof utilizes the Poincar\'e hyperbolic metric in terms of which both metrics can be expressed. This identity then yields sharp distortion results for quasiregular mappings and analytic functions, expressed in terms of the Hilbert metric. We also prove that Hilbert circles are, in fact, Euclidean ellipses. The proof makes use of computer algebra methods. In particular, Gr\"obner bases are used.

math.CV↗
arXiv2024

Visual angle metric in the upper half plane

We prove an identity which connects the visual angle metric $v_{\mathbb{H}^2}$ and the hyperbolic metric $\rho_{\mathbb{H}^2}$ of the upper half plane $\mathbb{H}^2$. The proof is based on geometric arguments and uses computer algebra methods for formula manipulation. We also prove a sharp H\"older continuity result for quasiregular mappings with respect to the visual angle metric.

math.MG↗
arXiv2023

Formulas for the visual angle metric

We prove several new formulas for the visual angle metric of the unit disk in terms of the hyperbolic metric and apply these to prove a sharp Schwarz lemma for the visual angle metric under quasiregular mappings.

math.CV↗
arXiv2022

Collinearity of points on Poincar\'e unit disk and Riemann sphere

We study certain points significant for the hyperbolic geometry of the unit disk. We give explicit formulas for the intersection points of the Euclidean lines and the stereographic projections of great circles of the Riemann sphere passing through these points. We prove several results related to collinearity of these intersection points, offer new ways to find the hyperbolic midpoint, and represent a formula for the chordal midpoint. The proofs utilize Gr\"obner bases from computer algebra for the solution of polynomial equations.

math.MG↗
arXiv2022

Geometric properties of Blaschke-like maps on domains with a conic boundary

For a circle $ C $ contained in the unit disk, the necessary and sufficient condition for the existence of a triangle inscribed in the unit circle and circumscribed about $ C $ is known as Chapple's formula. The geometric properties of Blaschke products of degree 3 given by Daepp et al. (2002) and Frantz (2004) allow us to extend Chapple's formula to the case of ellipses in the unit disk. The main aim of this paper is to provide a further extension of Chapple's formula. Introducing a Blaschke-like map of a domain whose boundary is a conic, we extend their results to the case where the outer curve is an ellipse or a parabola. Moreover, we also give some geometrical properties for the Blaschke-like maps of degree $ d $.

math.CV↗
arXiv2021

The Ptolemy-Alhazen problem and quadric surface mirror reflection

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse, and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors.

math.CV↗
arXiv2020

A new intrinsic metric and quasiregular maps

We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.

math.CV↗
arXiv2019

Barrlund's distance function and quasiconformal maps

Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ 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.

math.CV↗
arXiv2017

The Ptolemy-Alhazen problem and spherical mirror reflection

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.

math.CV↗
arXiv2014

Rational functions with nodes

A natural kind of compactification of the virtual moduli spaces of rational functions of one complex variable is given. To describe the boundary points geometrically, the authors introduce the concept of rational functions with nodes, defined on partially crushed punctured Riemann spheres with nodes.

math.CV↗