It is an easy exercise that using ruler and compass one find the focus of a given parabola.
Can one do the same using only a ruler? -- if not, why?
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$\begingroup$ Is this for a ruler or a straightedge and compass? $\endgroup$– Random832Commented Dec 31, 2014 at 20:22
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$\begingroup$ @Random832: yes, perhaps, but i don't know -- my native languages are Russian, Maths and "things-as-they-are". But if you talk about en.wikipedia.org/wiki/Straight_edge then i agree! $\endgroup$– VictorCommented Jan 1, 2015 at 6:13
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1 Answer
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No, because there exists a projective map which preserves the parabola but does not preserve its focus (this is so for any conic and any point.)
answered Dec 31, 2014 at 16:48
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$\begingroup$ Fantastic! It's very easy and hence beautiful and hence genius! Happy New Year $\endgroup$– VictorCommented Dec 31, 2014 at 17:38
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$\begingroup$ You need to define the focus and by doing so I guess you might already see that this is not projective. Also there are points that are always preserved by a projective transform, the point at infinity in the case of the parabola so this contradicts your final statement. Can you modify it to make it always hold? $\endgroup$ Commented Apr 28, 2016 at 7:39
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$\begingroup$ Assume that there is some construction on the plane $\pi$ which given parabola $\gamma$ gives its focus $F$. Let $T$ be projective map for which $T(\gamma)$ is a parabola again, but $T(F)$ is not it's focus. Apply the same construction to $T(\gamma)$ on $T(\pi)$, we get $T(F)$. Contradiction. $\endgroup$ Commented Apr 28, 2016 at 8:35