Geometry of Continued Fractions with Real Elements and Kepler’s Second Law

  • Oleg Karpenkov
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)


In the beginning of this book we discussed the geometric interpretation of regular continued fractions in terms of LLS sequences of sails. Is there a natural extension of this interpretation to the case of continued fractions with arbitrary elements? The aim of this chapter is to answer this question.

We start with a geometric interpretation of odd or infinite continued fractions with arbitrary elements in terms of broken lines in the plane having a selected point (say the origin). Further, we consider differentiable curves as infinitesimal broken lines to define analogues of continued fractions for curves. The resulting analogues possess an interesting interpretation in terms of a motion of a body according to Kepler’s second law.


Regular Continued Fraction Broken Line Interesting Interpretation Geometric Interpretation Angular Density 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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