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BMAD

Determining an ellipse given a foci

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BMAD    62

How many points would you need to have to uniquely determine an ellipse given that you know a foci is located at (0,0).

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gavinksong    11

Minor grammar correction: the singular form of foci is focus.

Without really thinking about it, I'm going to guess:

Spoiler

Two points. The intuition is that one point would contain less information than the minimal definition of an ellipse (minus a focus), whereas two points contains more.

 

Edited by gavinksong
wording

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gavinksong    11

After some more thought,

Spoiler

Three points. With each point, we get the equation:

R = |p| + |p - f|

where R and f are unknown values, and p is the new point in vector form.

There are three unknowns (since f is 2-dimensional), which we need at least three equations to solve.

 

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bonanova    76
Spoiler

Ellipse is the locus of points the sum of whose distances from the foci is constant.

So, we just need (a) the other locus and (b) any point on the ellipse to determine the sum: two more points.

 

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Spoiler

It's odd that given one focus, and two points on the ellipse, is not enough. But three points on the ellipse, with one focus, should be enough to determine the ellipse and the other focus. And four points on the ellipse, and no foci, would also work. But how to do this is beyond me.

 

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gavinksong    11
On 9/16/2017 at 12:56 AM, Quantum.Mechanic said:
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It's odd that given one focus, and two points on the ellipse, is not enough. But three points on the ellipse, with one focus, should be enough to determine the ellipse and the other focus. And four points on the ellipse, and no foci, would also work. But how to do this is beyond me.

 

My thoughts exactly... except is it possible with four points?

Edited by gavinksong

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