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<p>The eccentricity of the inner ellipse in a 3-Poncelet pair with the unit circle coincides with the pseudo-hyperbolic distance between its foci. Based on Barbilian’s metrization procedure, which employs Apollonian circles to express the pseudo-hyperbolic distance within the unit disk, a new proof of this fact is obtained together with a geometric construction of the corresponding ellipse. Two pairs of Apollonian circles, classical and dual, arise as natural geometric and metric guides, their inversive distance and intersection angle being directly related to the ellipse’s eccentricity. The methods are elementary and establish new connections among inversive geometry, hyperbolic geometry, and Poncelet configurations.</p>

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Inversive distance and Poncelet pairs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月13日

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  • Liliana Gabriela Gheorghe,
  • Ronaldo Alves Garcia

摘要

The eccentricity of the inner ellipse in a 3-Poncelet pair with the unit circle coincides with the pseudo-hyperbolic distance between its foci. Based on Barbilian’s metrization procedure, which employs Apollonian circles to express the pseudo-hyperbolic distance within the unit disk, a new proof of this fact is obtained together with a geometric construction of the corresponding ellipse. Two pairs of Apollonian circles, classical and dual, arise as natural geometric and metric guides, their inversive distance and intersection angle being directly related to the ellipse’s eccentricity. The methods are elementary and establish new connections among inversive geometry, hyperbolic geometry, and Poncelet configurations.