Can someone prove that the minimum distance between two non-intersecting parabolas is along their common normal (without calculus)? (If I understand how to prove it, I'll prove it myself for other curves too.)
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J. M. ain't a mathematician
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Archer
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I believe you must use calculus to prove it. See https://math.stackexchange.com/questions/1209893/minimum-distance-between-two-parabolas – Landuros Oct 23 '17 at 10:25
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1This is not true if the parabolas intersect. – Mark Bennet Oct 23 '17 at 10:43
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Or if they're nested like $y=x^2$ and $y=x^2+1$. Note that in this case a shortest connection does not exist in the (Euclidean) plane. – Oscar Lanzi Oct 23 '17 at 12:31
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Assuming the existence of a shortest distance, let the shortest distance join points $A$ and $B$ and construct a circle with diameter $AB$. If the circle cuts either parabola at a point other than $A$ or $B$ there is a line joining the two curves which is shorter than the diameter. Else the circle must be tangent at points $A$ and $B$ and the diameter $AB$ is normal to both curves (because the diameter is perpendicular to the tangent).
There are some implicit assumptions here, but would this do? Note that if $A=B$ then the diameter doesn't have a fixed direction and the situation is different.
Mark Bennet
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1@Raffaele No - I assume their existence (see very first comment in answer) and show that if they exist they are joined by a common normal. (Implicit assumptions include that the parabolas do not cross). If there is a shortest distance, there are points which are that distance apart. – Mark Bennet Oct 23 '17 at 14:01