On asymmetry axes of the Oval of Descartes

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  • デカルトの卵形線の内外分枝の非対称軸について
  • デカルト ノ ランケイセン ノ ナイガイ ブン シ ノ ヒタイショウジク ニ ツイテ

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Abstract

We define the inner (+) and the outer (-) part of the Cartesian Oval as mr1±nr2 = kc on bipolar coordinates.<BR>We can consider on Minor axis (asymmetry axis) of the inner part of the Oval, and can define Major axis (asymmetry axis) of the outer part of the Oval. This major axis is a segment which connects the middle point O of symmetry axis and the point Fp on the oval, which is at the longest distance from the point O. Then, the length of major axis is ao (1 + eL eR) 1/2 (where ao is a half of the length of the symmetry-axis, eL, eR are left and right eccentricity of the Oval, respectively.) And, we can say that Cardioid is the special case of Cartesian Oval. In this case, eL and eR are equal to 1 and the length of the major axis is ao 21/2.<BR>Moreover, we have found the following Lemma. [Lemma] Let bi be the length of Minor axis of the inner part of the Oval, let ai and ao be the half length of symmetry-axis of the inner and outer part, respectively. Let bo be the length of the Major axis of the outer part. Then, the following invariant holds. (bi/ai) 2+ (bo/ao) 2=2

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