Closest distance between two conics (ellipse,hyp.,par.)

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The discussion focuses on calculating the closest distance between two ellipses in space, defined by their respective orbital equations and Euler angles. Participants seek both symbolic solutions and approximation techniques for this problem, particularly when dealing with non-closed orbits like hyperbolas and parabolas. The challenge includes finding a formula for the distance between two quadratic figures in higher dimensions, which appears to be underexplored in existing literature. The conversation emphasizes the need for mathematical frameworks to address these complex geometric relationships. Overall, the thread highlights a significant gap in available solutions for calculating distances between conic sections in various orbital configurations.
petra
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I have question for you,How calculate closest distance between two
ellipse in space.
orbit first : r = (p1)/(1+epsilon1*cos(theta1))

second orbit : r = (p2)/(1+epsilon2*cos(theta2))

the relation between two ellipse is some euler angles call them
first angle :a1
second angle :b1 (these three euler angles)
third angle :c1

(taking in account they have same focus :the sun)

? symbolic solution for this problem
? how solve this with approximation techniques.


what when orbit is not closed:hyperbole,parabole
 
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I have the same type of problem and I am looking for a formula for the distance between two quadratic figures in \Re^n but I haven't seen it anywhere.
 

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