For any conic section, the eccentricity is the ratio of the distance to the focus and directrix. If the eccentricity were defined to be negative, would this have any significant meaning/application, or potentially be linked to complex numbers?
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The excentricity of conic sections is thoroughly explained in : Confusion with the various forms of the equation of second degree . I really don't see how to add more to it. Not everything can be "generalized" in a sensible manner. Good question (+1) though. – Han de Bruijn Feb 17 '17 at 16:27
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It actually doesn't change anything; if you look at the equations for finding the eccentricity, it is the square root of something, and since there are two square roots of a number, it can be both negative or positive.
If you make a plot of a conic section based on the eccentricity, it doesn't change at all. Plot it on desmos graphing calculator using the formula r=$\frac{d}{1-dcos(\theta)}$ where d is the eccentricity (polar coordinates). Nothing happens if you negate the eccentricity.
Jonathan
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We've been studying this exact question for all of 5 minutes. We have an answer. No.
This is why: The formula for eccentricity is $c/a$ which is the distance from the center to a foci divided by the distance from the center to a vertex. If eccentricity were negative, either the numerator or denominator would have to be negative. It is IMPOSSIBLE to have a negative distance for $a$ or $c$. Therefore, eccentricity cannot be negative.
Xander Henderson
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I have no problem with the notion of a signed distance, which represents distance in a particular direction. That may not be entirely relevant here, but it is clearly not "IMPOSSIBLE. I am also curious to know what you think of the other answer provided here, which has no difficulty with the concept of a negative distance or eccentricity. – Xander Henderson Nov 12 '19 at 19:43