Symmetrick-ellipses with limiting eccentricity

Abstract

Consider a sequence of ellipses in which each new member is an ellipse which has as foci the two points of intersection of the previous ellipse with its minor axis and which passes through the two points of intersection of that ellipse with its major axis. Kapur [4] has shown that the eccentricity of the ellipses in this sequence approaches the value \(2/(\sqrt 5 + 1) \simeq 0.6103\) which is the reciprocal of the so-called Golden Ratio. In this paper we show that for any finite integerk ≥ 2 we can construct a sequence of symmetrick-ellipses such that the eccentricity of the ellipses in the sequence approaches a limite_k for eachk. We also show that for these sequencese_k approaches the limit π/4 ask approaches infinity. In conclusion we present some computed approximations fore_k for various values ofk and indicate how the computations were performed.