How is the critical parameter Eccentricity ($e$) mathematically defined in relation to the ellipse geometry?

Answer

It is the distance from the center to a focus divided by the semi-major axis ($e = c/a$)

Eccentricity ($e$) serves as the quantitative measure describing how much the elliptical path deviates from a perfect circle. Geometrically, this is derived by relating the linear dimension defining the focus position to the overall size of the ellipse. Specifically, $c$ represents the distance from the geometric center of the ellipse to either focus, and $a$ represents the semi-major axis, which is half the longest diameter. The ratio $c/a$ directly quantifies this deviation. A circle has $c=0$, yielding $e=0$. An increase in $e$ means $c$ approaches $a$, resulting in a highly elongated shape where the foci are spread far apart relative to the overall path length.

How is the critical parameter Eccentricity ($e$) mathematically defined in relation to the ellipse geometry?

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