Minimum distance between ellipses

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SUMMARY

The discussion focuses on calculating the minimum distance between two non-interacting bodies on elliptical orbits, specifically addressing the mathematical minimization of a function of two variables within the domain [0,2*pi)*[0,2*pi). The user suggests utilizing numerical methods, such as the conjugate gradient method, while also inquiring about potential analytical approaches that could simplify the problem. The context involves two ellipses representing the orbits of minor planets in 3D space, with the assumption that they share one of their foci.

PREREQUISITES
  • Understanding of elliptical orbits in celestial mechanics
  • Familiarity with numerical optimization techniques, specifically the conjugate gradient method
  • Basic knowledge of minimization problems in multivariable calculus
  • Concept of foci in ellipses and their geometric implications
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  • Research analytical methods for minimizing functions of two variables
  • Explore advanced numerical techniques for optimization in 3D space
  • Study the properties of ellipses and their foci in celestial mechanics
  • Investigate existing algorithms for distance calculations between parametric curves
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Astronomers, mathematicians, and physicists involved in celestial mechanics, as well as computational scientists working on orbital simulations and distance calculations between celestial bodies.

Coelum
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I'm addressing the problem of computing the minimum possible distance between two non-interacting bodies on elliptical orbits. From a general point of view, it looks like a minimization problem of a function of two variables, e.g. in the domain [0,2*pi)*[0,2*pi). This problem can be numerically addressed in a standard fashion, e.g. by a conjugate gradient method. But I wonder if an analytical approach exists that can simplify the problem - maybe reducing it to unidimensional - and significantly speed-up the computation.
I have posted the some question on the Celestial Mechanics Forum - no reply so far.
 
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If the path is ellptical, wouldn't it just be a straight line representing the minor axis of the ellipse?
 
I guess I've not been clear enough: I'm dealing with two ellipses (e.g., representing the paths of two minor planets around the sun) in 3D. Of course, you may assume that they share one of the foci.
 

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