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Subject: "product of distances"     Previous Topic | Next Topic
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ociem
Member since Aug-11-07
Aug-11-07, 06:44 AM (EST)
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"product of distances"
 
   If you take two fixed points and then plot all the points where the two distances sum to a constant you get an ellipse. If you do a subtraction, you get a hyperbola. Thanks to this wonderful site, I now know that if you fix the ratio of these distances, you get a circle. my question is what do you get if you make the product of these distances a constant? I can't be the first one who has wondered about this, surely :)


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mpdlc
guest
Aug-11-07, 02:08 PM (EST)
 
1. "RE: product of distances"
In response to message #0
 
   The curve is the

Lemniscate of Bernoulli

If interested you can check among others the links below

https://mathworld.wolfram.com/Lemniscate.html
https://xahlee.org/SpecialPlaneCurves_dir/LemniscateOfBernoulli_dir/lemniscateOfBernoulli.html


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ociem
guest
Aug-12-07, 01:29 AM (EST)
 
2. "RE: product of distances"
In response to message #1
 
   Thanks. I think the lemniscate may be a special case of this. The wolfram site shows it as a cross section of a torus. This makes sense based on what I had found imperically. I plotted one of these curves, but it was one continuous figure. It must have been closer to the edge of the torus. This is a great site. I'm sure I'll be back.


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mpdlc
guest
Aug-12-07, 06:55 AM (EST)
 
3. "RE: product of distances"
In response to message #2
 
   Actually the lemniscate is a particular case of Cassini Ovals.
It is easy to write the general analytical equation for these curves assuming both points C1 and C2 on X axis at same distance of origin O. Then their coordinates are C1 (c, 0) and C2 (-c, 0), calling just for convenience the produt of distance a^4. It can be write immediately the equation expressing the constant product of the distance to them from a generic point P(x,y)

((x-c)^2 + y^2) ((x+c)^2 + y^2) = a^4.

If c = a we have the classic lemniscate of Bernoulli, otherwise we obtain the Ovals that probably will explain your other results


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mpdlc
guest
Aug-12-07, 06:55 AM (EST)
 
4. "RE: product of distances"
In response to message #2
 
   Below the link for the Cassini Ovals I forgot to include in my post

https://mathworld.wolfram.com/CassiniOvals.html


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