# Max & min distance between two moving points

I am trying to find the minimal and maximum distance between two moving points with Mathematica in an interval.

For example,

p1 = (Sin(t), Cos(t))
p2 = (Cos(3t), Sin(2t))


Interval is $0≤t≤2\pi$.

Can I use the command EuclideanDistance or should I use FindMinimum, FindMaximum, NMinimize, Maximize; and how do I use these commands in an interval?

• Please, consider updating your question to include what you have tried and where you are getting stuck. That way, people on this site will know exactly what help you need.
– user9660
Feb 25, 2016 at 18:55
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– user9660
Feb 25, 2016 at 18:55

p1 = {Sin[t], Cos[t]}; p2 = {Cos[3 t], Sin[2 t]};

tAtMin = ArgMax[{EuclideanDistance[p1, p2]^2, 0 <= t <= 2 Pi}, t]
tAtMax = ArgMin[{EuclideanDistance[p1, p2]^2, 0 <= t <= 2 Pi}, t]

ParametricPlot[{p1, p2}, {t, 0, 2 Pi},
Epilog -> {PointSize[0.02],
Red, Thick, Dashed, Through[{Point, Line}[{p1, p2} /. t -> tAtMax]],
Darker@Green, Thick, Dashed, Through[{Point, Line}[{p1, p2} /. t -> tAtMin]]
}
]


Here's a way to visualize the evolution of the distance as $t$ varies:

Animate[
Show[{
ParametricPlot[{p1, p2}, {t, 0, 2 Pi},
Epilog -> {PointSize[0.02],
Red, Thick, Dashed,
Through[{Point, Line}[{p1, p2} /. t -> tAtMax]],
Darker@Green, Thick, Dashed, Through[{Point, Line}[{p1, p2} /. t -> tAtMin]]
}
],
Graphics[{
Thick, Gray, Dashed, PointSize[0.02],
Through[{Point, Line}[{{Sin[x], Cos[x]}, {Cos[3 x], Sin[2 x]}}]]
}]
}],
{x, 0, 2 Pi},
AnimationRate -> .05
]


• Wow, what an amazing answer! Feb 25, 2016 at 18:38
• @C.Woods Thank you! Feb 25, 2016 at 18:47
• Wow. Amazing, much more than I expected. Thank you very much!
– DCB
Feb 26, 2016 at 0:02
• @DCB I'm very glad I could help Feb 26, 2016 at 4:46
• The use of Through[{Point, Line}[{p1, p2} /. t -> tAtMax]] to create simultaneously the two points and the line joining them is particularly clever (sneaky?). A tip (trick?) well worth remembering! Feb 26, 2016 at 15:31