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For example in the arrow shown in the photo above, assuming the two ellipses have the same eccentricity, will that distance be the same between any two parallel points between the two ellipses?

tinlyx
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Sofia
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    If you mean the vertical distance, the answer is obvious from the figure. –  May 27 '19 at 07:28

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The locus of the points at constant Euclidean distance of an ellipse (also called an offset curve) is not an ellipse. This answers you question more generally.