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Inscribed Ellipses and Blaschke Products

Inscribed Ellipses and Blaschke Products A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of a bicentric triangle or quadrilateral is known as Chapple’s formula or Fuss’ formula, respectively. In this paper, we give natural extensions of these formulae. For an ellipse $$ E $$ E , the following two conditions are equivalent: (1) there exists a triangle and a quadrilateral, which $$ E $$ E is inscribed in and the unit circle is circumscribed about, respectively; (2) for some $$ a,b $$ a , b in the unit disk, $$ E $$ E is defined by the equation $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | or $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \sqrt{(|a|^2+|b|^2-2)/(|a|^2|b|^2-1)}, \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | ( | a | 2 + | b | 2 - 2 ) / ( | a | 2 | b | 2 - 1 ) , respectively. To prove them, we use some geometrical properties of Blaschke products on the unit disk, which also give new proofs of Chapple’s and Fuss’ formulae. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Inscribed Ellipses and Blaschke Products

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Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-013-0037-8
Publisher site
See Article on Publisher Site

Abstract

A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of a bicentric triangle or quadrilateral is known as Chapple’s formula or Fuss’ formula, respectively. In this paper, we give natural extensions of these formulae. For an ellipse $$ E $$ E , the following two conditions are equivalent: (1) there exists a triangle and a quadrilateral, which $$ E $$ E is inscribed in and the unit circle is circumscribed about, respectively; (2) for some $$ a,b $$ a , b in the unit disk, $$ E $$ E is defined by the equation $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | or $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \sqrt{(|a|^2+|b|^2-2)/(|a|^2|b|^2-1)}, \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | ( | a | 2 + | b | 2 - 2 ) / ( | a | 2 | b | 2 - 1 ) , respectively. To prove them, we use some geometrical properties of Blaschke products on the unit disk, which also give new proofs of Chapple’s and Fuss’ formulae.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 26, 2013

References