Minimum Distance Between Earth & Mars: Analytic Calc

[deltasata]

What is the minimum distance between Earth and Mars in course of their rotation in orbits?
I need the derivation and analytic calculation.
If we assume the plane of rotation is different for Earth and mars, the problem become to complicated. So the orbits may be assumed co-planner.Please provide me some useful links...

 

[mfb]

The minimal distance when? Within the next 100 years, or within geological timescales?

55.7 million km

You get a rough approximation by subtracting mars' perihelion and Earth's semi-major axis:
Earth 149,598,261 km
Mars 206,669,000 km
Difference 57 million km.

A better value would require to look at Earth's eccentricity and the relative orientation of the ellipses.