Almost closed interscribed polygons
  We demonstrate a new approach to the computation of ratios of elliptic integrals. It turns out that almost closed polygons interscribed between two conics retain some of the properties of such closed polygons. We apply these retained properties to compute ratios of an incomplete elliptic integral over the complete one. This computation is based on an iterative procedure to determine the sequence of vertices of a polygon interscribed between two conics. Surprisingly, some iteration numbers are the denominators of the convergent fractions for thye ratio of some elliptic integrals. The algorithm ensures high precision, is numerically stable and as fast as the arithmetic-geometric mean method, though not faster. Nonetheless, there are reasons to consider the proposed algorithm as it is quite differentfrom other integration methods and may be applicable to other problems. 
Yury Kroll and Boris Mirman
math.DS