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2 years ago in Computational Geometry By Saqib

How can the length of an ellipse or elliptical arc be approximated?

The exact arc length of an ellipse involves elliptic integrals of the second kind, which aren’t elementary. In my computational physics projects, I need a balance of speed and accuracy. I’m looking for advice on which approximation formulas or numerical quadrature methods are most robust for different eccentricities, and how to implement them effectively without resorting to heavy symbolic libraries.

length EllipticIntegral ArcLength

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By Pravin Patel Answered 2 years ago

Having implemented these for orbital mechanics simulations, I can tell you the choice depends on your required precision and eccentricity. For most engineering purposes, I recommend Ramanujan's second approximation for the circumference it’s astonishingly accurate and simple to code. For a specific arc segment, avoid naive discrete summation. Instead, use a dedicated numerical integrator like Gaussian quadrature on the parametric integral. I have seen 16-point Gauss-Legendre quadrature deliver excellent results even for highly eccentric ellipses. For repeated calls, pre-computing a lookup table parameterized by eccentricity and angle is the most efficient approach.

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