What question did this study set out to answer?

This work aims to establish new algorithms for calculating ellipse perimeter with higher accuracy and lower complexity.

May 8, 2026Open Access

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

Key Points

This work aims to establish new algorithms for calculating ellipse perimeter with higher accuracy and lower complexity.
Utilized the Circle-Distance Identity to relate ellipse speed and perimeter through circular-distance laws.
Integrated Mahāvīra’s historical insights into a modern minimax framework for analysis.
Derived the mūla-bheda coordinate for more efficient algebraic evaluations.
Achieved an accuracy-complexity improvement factor of approximately 5.5 million over the classical Ramanujan II benchmark.
Demonstrated that the new approach offers a more compact and analytic means of evaluating ellipse perimeter.

Abstract

The Bhāradvāja mūla-bheda heads establish a new accuracy–complexity bound for ellipse perimeter geometry, outperforming the classical Ramanujan II benchmark by a factor of approximately 5.5 million on the stated benchmark domain. This result is driven by the Circle-Distance Identity, which represents ellipse speed as a pointwise circular-distance law and the perimeter as a mean circular-distance law. By bringing Mahāvīra’s 9th-century āyata-vṛtta endpoint intuition into a modern minimax framework, this work identifies the mūla-bheda coordinate as a natural analytic basis for compact algebraic evaluation.

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Authors

Rāma Bhāradvāja

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Institutions

Indian Institute of Technology Madras

Universitas Wiraraja

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

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Abstract

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