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Concyclic Intervals in the Plane

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Abstract

We study arrangements of intervals in \({\mathbb {R}}^2\) for which many pairs are concyclic. We show that any set of intervals with many concyclic pairs must have underlying algebraic and geometric structure. In the most general case, we prove that the endpoints of many intervals belong to a single bicircular quartic curve.

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Notes

  1. A circle is ordinary if it contains exactly three points of P.

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Acknowledgements

The authors thank Daniel Di Benedetto for valuable suggestions and conversations throughout the preparation of this paper. The research of the first author was supported in part by an NSERC Discovery grant and OTKA K 119528 grant. The research of the second author was supported in part by Killam and NSERC doctoral scholarships.

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Authors and Affiliations

  1. The University of British Columbia, Vancouver, Canada

    József Solymosi & Ethan Patrick White

  2. Obuda University, Budapest, Hungary

    József Solymosi

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  1. József Solymosi
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  2. Ethan Patrick White
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Correspondence to Ethan Patrick White.

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Editor in Charge: Csaba D. Tóth

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Appendix

Appendix

Below we give the bicircular quartic curve \(P(a,b)=0\) discussed in Sect. 4.3. The constant term is 0. The degree four homogeneous component:

$$\begin{aligned}{} & {} (a^2+b^2)^2(a_1d_1b_2-a_1d_1d_2-b_1c_1b_2+b_1c_1d_2\\{} & {} \qquad \qquad \qquad \qquad \qquad -\,b_1a_2d_2+b_1b_2c_2+d_1a_2d_2-d_1b_2c_2). \end{aligned}$$

Degree three homogeneous component:

$$\begin{aligned}&(a^2+b^2)(a_1^2c_1b_2b-a_1^2c_1d_2b-a_1^2d_1b_2a +a_1^2d_1d_2a+a_1^2a_2d_2b-a_1^2b_2c_2b\\&\quad -a_1c_1^2b_2b + a_1c_1^2d_2b -a_1d_1^2b_2b+a_1d_1^2d_2b-a_1d_1a_2^2b +a_1d_1a_2b-a_1d_1b_2^2b\\&\quad -a_1d_1b_2a + a_1d_1c_2^2b - a_1d_1c_2b +a_1d_1d_2^2b+a_1d_1d_2a-a_1a_2d_2b+a_1b_2c_2b\\&\quad +b_1^2c_1b_2b-b_1^2c_1d_2b-b_1^2d_1b_2a + b_1^2d_1d_2a +b_1^2a_2d_2b-b_1^2b_2c_2b+b_1c_1^2b_2a\\&\quad -b_1c_1^2d_2a+b_1c_1a_2^2b - b_1c_1a_2b + b_1c_1b_2^2b + b_1c_1b_2a -b_1c_1c_2^2b+b_1c_1c_2b \\&\quad -b_1c_1d_2^2b-b_1c_1d_2a +b_1d_1^2b_2a-b_1d_1^2d_2a - b_1a_2^2c_2b + b_1a_2^2d_2a +b_1a_2c_2^2b\\&\quad +b_1a_2d_2^2b+b_1a_2d_2a -b_1b_2^2c_2b+b_1b_2^2d_2a-b_1b_2c_2^2a-b_1b_2c_2a - b_1b_2d_2^2a \\&\quad -c_1^2a_2d_2b+c_1^2b_2c_2b+c_1a_2d_2b-c_1b_2c_2b -d_1^2a_2d_2b+d_1^2b_2c_2b+d_1a_2^2c_2b \\&\quad - d_1a_2^2d_2a -d_1a_2c_2^2b-d_1a_2d_2^2b-d_1a_2d_2a+d_1b_2^2c_2b -d_1b_2^2d_2a + d_1b_2c_2^2a \\&\quad + d_1b_2c_2a + d_1b_2d_2^2a). \end{aligned}$$

Degree two homogeneous component:

$$\begin{aligned}&-a_1^2c_1a_2^2b^2 + a_1^2c_1a_2b^2 - a_1^2c_1b_2^2b^2 - a_1^2c_1b_2ab + a_1^2c_1c_2^2b^2 \\&\quad - a_1^2c_1c_2b^2 + a_1^2c_1d_2^2b^2 + a_1^2c_1d_2ab + a_1^2d_1a_2^2ab - a_1^2d_1a_2d_2a^2 \\&\quad - a_1^2d_1a_2d_2b^2 - a_1^2d_1a_2ab + a_1^2d_1b_2^2ab + a_1^2d_1b_2c_2a^2 + a_1^2d_1b_2c_2b^2\\&\quad + a_1^2d_1b_2a^2 - a_1^2d_1c_2^2ab + a_1^2d_1c_2ab - a_1^2d_1d_2^2ab - a_1^2d_1d_2a^2 \\&\quad + a_1^2a_2^2c_2b^2 - a_1^2a_2^2d_2ab - a_1^2a_2c_2^2b^2 - a_1^2a_2d_2^2b^2 + a_1^2b_2^2c_2b^2 \\&\quad - a_1^2b_2^2d_2ab + a_1^2b_2c_2^2ab + a_1^2b_2d_2^2ab + a_1c_1^2a_2^2b^2 - a_1c_1^2a_2b^2\\&\quad +a_1c_1^2b_2^2b^2 + a_1c_1^2b_2ab - a_1c_1^2c_2^2b^2 + a_1c_1^2c_2b^2 - a_1c_1^2d_2^2b^2 \\&\quad - a_1c_1^2d_2ab + a_1d_1^2a_2^2b^2 - a_1d_1^2a_2b^2 +a_1d_1^2b_2^2b^2+a_1d_1^2b_2ab\\&\quad - a_1d_1^2c_2^2b^2 + a_1d_1^2c_2b^2 - a_1d_1^2d_2^2b^2 - a_1d_1^2d_2ab + a_1d_1a_2^2d_2a^2 \\&\quad + a_1d_1a_2^2d_2b^2 + a_1d_1b_2^2d_2a^2 + a_1d_1b_2^2d_2b^2 - a_1d_1b_2c_2^2a^2 - a_1d_1b_2c_2^2b^2 \\&\quad - a_1d_1b_2d_2^2a^2 - a_1d_1b_2d_2^2b^2 - a_1a_2^2c_2b^2 + a_1a_2^2d_2ab + a_1a_2c_2^2b^2 \\&\quad + a_1a_2d_2^2b^2 - a_1b_2^2c_2b^2 + a_1b_2^2d_2ab - a_1b_2c_2^2ab - a_1b_2d_2^2ab \\&\quad - b_1^2c_1a_2^2b^2 + b_1^2c_1a_2b^2 - b_1^2c_1b_2^2b^2 - b_1^2c_1b_2ab + b_1^2c_1c_2^2b^2 \\&\quad - b_1^2c_1c_2b^2 + b_1^2c_1d_2^2b^2 + b_1^2c_1d_2ab + b_1^2d_1a_2^2ab - b_1^2d_1a_2d_2a^2 \\&\quad - b_1^2d_1a_2d_2b^2 - b_1^2d_1a_2ab + b_1^2d_1b_2^2ab + b_1^2d_1b_2c_2a^2 + b_1^2d_1b_2c_2b^2 \\&\quad + b_1^2d_1b_2a^2 -b_1^2d_1c_2^2ab+b_1^2d_1c_2ab - b_1^2d_1d_2^2ab - b_1^2d_1d_2a^2 \\&\quad + b_1^2a_2^2c_2b^2 - b_1^2a_2^2d_2ab - b_1^2a_2c_2^2b^2 - b_1^2a_2d_2^2b^2 + b_1^2b_2^2c_2b^2 \\&\quad - b_1^2b_2^2d_2ab + b_1^2b_2c_2^2ab + b_1^2b_2d_2^2ab - b_1c_1^2a_2^2ab + b_1c_1^2a_2d_2a^2 \\&\quad + b_1c_1^2a_2d_2b^2 + b_1c_1^2a_2ab - b_1c_1^2b_2^2ab - b_1c_1^2b_2c_2a^2 - b_1c_1^2b_2c_2b^2 \\&\quad - b_1c_1^2b_2a^2 + b_1c_1^2c_2^2ab - b_1c_1^2c_2ab + b_1c_1^2d_2^2ab + b_1c_1^2d_2a^2 \\&\quad - b_1c_1a_2^2d_2a^2 - b_1c_1a_2^2d_2b^2 - b_1c_1b_2^2d_2a^2 - b_1c_1b_2^2d_2b^2 + b_1c_1b_2c_2^2a^2 \\&\quad + b_1c_1b_2c_2^2b^2 + b_1c_1b_2d_2^2a^2 + b_1c_1b_2d_2^2b^2 - b_1d_1^2a_2^2ab + b_1d_1^2a_2d_2a^2\\&\quad + b_1d_1^2a_2d_2b^2 + b_1d_1^2a_2ab - b_1d_1^2b_2^2ab - b_1d_1^2b_2c_2a^2 - b_1d_1^2b_2c_2b^2 \\&\quad - b_1d_1^2b_2a^2 + b_1d_1^2c_2^2ab - b_1d_1^2c_2ab + b_1d_1^2d_2^2ab + b_1d_1^2d_2a^2 \\&\quad + b_1a_2^2c_2ab - b_1a_2^2d_2a^2 - b_1a_2c_2^2ab - b_1a_2d_2^2ab + b_1b_2^2c_2ab\\&\quad - b_1b_2^2d_2a^2 + b_1b_2c_2^2a^2 + b_1b_2d_2^2a^2 - c_1^2a_2^2c_2b^2 + c_1^2a_2^2d_2ab\\&\quad + c_1^2a_2c_2^2b^2 + c_1^2a_2d_2^2b^2 - c_1^2b_2^2c_2b^2 + c_1^2b_2^2d_2ab - c_1^2b_2c_2^2ab \\&\quad - c_1^2b_2d_2^2ab + c_1a_2^2c_2b^2 - c_1a_2^2d_2ab - c_1a_2c_2^2b^2 - c_1a_2d_2^2b^2\\&\quad + c_1b_2^2c_2b^2 - c_1b_2^2d_2ab + c_1b_2c_2^2ab + c_1b_2d_2^2ab - d_1^2a_2^2c_2b^2\\&\quad + d_1^2a_2^2d_2ab + d_1^2a_2c_2^2b^2 + d_1^2a_2d_2^2b^2 - d_1^2b_2^2c_2b^2 + d_1^2b_2^2d_2ab \\&\quad - d_1^2b_2c_2^2ab - d_1^2b_2d_2^2ab - d_1a_2^2c_2ab + d_1a_2^2d_2a^2 + d_1a_2c_2^2ab \\&\quad + d_1a_2d_2^2ab - d_1b_2^2c_2ab + d_1b_2^2d_2a^2 - d_1b_2c_2^2a^2 - d_1b_2d_2^2a^2. \end{aligned}$$

Degree one homogeneous component:

$$\begin{aligned}&a_1^2c_1a_2^2d_2b - a_1^2c_1a_2d_2b + a_1^2c_1b_2^2d_2b - a_1^2c_1b_2c_2^2b + a_1^2c_1b_2c_2b\\&\quad - a_1^2c_1b_2d_2^2b - a_1^2d_1a_2^2c_2b + a_1^2d_1a_2c_2^2b + a_1^2d_1a_2d_2^2b + a_1^2d_1a_2d_2a \\&\quad - a_1^2d_1b_2^2c_2b - a_1^2d_1b_2c_2a - a_1c_1^2a_2^2d_2b + a_1c_1^2a_2d_2b - a_1c_1^2b_2^2d_2b \\&\quad + a_1c_1^2b_2c_2^2b - a_1c_1^2b_2c_2b + a_1c_1^2b_2d_2^2b - a_1d_1^2a_2^2d_2b + a_1d_1^2a_2d_2b\\&\quad - a_1d_1^2b_2^2d_2b + a_1d_1^2b_2c_2^2b - a_1d_1^2b_2c_2b + a_1d_1^2b_2d_2^2b + a_1d_1a_2^2c_2b \\&\quad - a_1d_1a_2^2d_2a - a_1d_1a_2c_2^2b - a_1d_1a_2d_2^2b + a_1d_1b_2^2c_2b-a_1d_1b_2^2d_2a\\&\quad +a_1d_1b_2c_2^2a+a_1d_1b_2d_2^2a+b_1^2c_1a_2^2d_2b - b_1^2c_1a_2d_2b + b_1^2c_1b_2^2d_2b \\&\quad - b_1^2c_1b_2c_2^2b + b_1^2c_1b_2c_2b - b_1^2c_1b_2d_2^2b - b_1^2d_1a_2^2c_2b + b_1^2d_1a_2c_2^2b\\&\quad + b_1^2d_1a_2d_2^2b + b_1^2d_1a_2d_2a\\&\quad - b_1^2d_1b_2^2c_2b - b_1^2d_1b_2c_2a + b_1c_1^2a_2^2c_2b - b_1c_1^2a_2c_2^2b - b_1c_1^2a_2d_2^2b\\&\quad - b_1c_1^2a_2d_2a + b_1c_1^2b_2^2c_2b +b_1c_1^2b_2c_2a-b_1c_1a_2^2c_2b + b_1c_1a_2^2d_2a\\&\quad + b_1c_1a_2c_2^2b + b_1c_1a_2d_2^2b - b_1c_1b_2^2c_2b + b_1c_1b_2^2d_2a - b_1c_1b_2c_2^2a\\&\quad -b_1c_1b_2d_2^2a+b_1d_1^2a_2^2c_2b - b_1d_1^2a_2c_2^2b - b_1d_1^2a_2d_2^2b - b_1d_1^2a_2d_2a \\&\quad + b_1d_1^2b_2^2c_2b + b_1d_1^2b_2c_2a. \end{aligned}$$

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Solymosi, J., White, E.P. Concyclic Intervals in the Plane. Discrete Comput Geom (2023). https://doi.org/10.1007/s00454-023-00515-y

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