Bhāradvāja Mūla-Bheda Jyāmiti: Algebraic Ellipse Perimeter Formulae and the Circle-Distance Identity

This paper recasts the ellipse perimeter as a problem in circle geometry.
The central result is the Circle-Distance Identity. For semi-axes a ≥ b, with m = (a+b)/2 and t = (a−b)/(a+b),
P(a, b) = 2πm·K(t),
where K(t) is the mean, around the circle, of the distance from an offset point (t, 0) to the unit circle. The ellipse perimeter is therefore exactly a mean circular distance, and the classical elliptic integral is the analytic shadow of that simpler law.

From it follows the Bhāradvāja mūla-bheda coordinate, q = (√a − √b)/(√a + √b), in which the exact perimeter becomes a positive central-binomial series and both endpoints remain exact. The coordinate yields a family of compact, non-iterative algebraic formulas, exact at both ends by construction, reaching leading accuracy among the compact non-iterative formulas tested. The reported maxima are peak-refined numerical values; no global interval bound is claimed.

The path to it ran through Mahāvīra's 9th-century rule for the ellipse, exact at the flat limit and, on the full benchmark, sharper there than the much later formulas of Kepler and Euler. Reconstructing that rule exposed a hidden circle constant, √10, the thread that led to the coordinate.

Included: the full paper and derivations, coefficient registries (degrees 2 to 12), a reproducible benchmark suite, and a Python (mpmath, NumPy) implementation.