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When reading about why no Venn diagram for four sets can be formed by intersecting four circles, I found that the author claimed that any two distinct circles can intersect in at most two points, while any two distinct ellipses can intersect in at most four.

Why is this? I can easily see from examples that it's intuitively obvious, but is there a geometric reason for it?

Thanks!

Community
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templatetypedef
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3 Answers3

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You know the fact that any circle can be characterised by any three points it passes through, i.e, If a circle passes through 3 points then it is unique. Therefore two distinct circles can intersect at no more than 2 points.

In case of ellipses, they are characterised by five points.

brainjam
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Phani Raj
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    Surely you mean five points for ellipses? If an ellipse was characterized by four points, then two distinct ellipses could not have four points in common. –  Oct 05 '12 at 05:33
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With the right viewpoint, any two distinct circles intersect in exactly four points, indeed any two distinct nondegenerate conics intersect in exactly four points. Like the circle of radius $2$ and the hyperbola $xy=1$. The relevant fact is the Theorem of Bézout, which says that under certain conditions of nondegeneracy, two plane curves, of degrees $m$ and $n$ respectively, intersect in $mn$ points, as long as you count multiplicity, set the points in the projective plane, and look for coordinates of intersection-points in an algebraically closed field. The equation of the circle $(x-1)^2 +y^2=1$, for instance, becomes $X^2-2XZ+Z^2+Y^2=Z^2$ upon homogenization, and it has the two distinct (in projective plane) points $(1,i,0)$ and $(1,-i,0)$. These are the two “imaginary points at infinity” carried by the circle, and they are common to all circles.

Two concentric circles intersect at these two points, and are tangent at both, so each point must be counted with multiplicity $2$.

Lubin
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It is because circles have a constant diameter. What that means is that if you have two circles intersecting, they cannot cross each other completely without fully overlapping. Whereas when using ellipses, there is a minor axis and a major axis. Which means that one side is shorter than the other, which means that one can completely cross the other. This produces 4 points of intersection, 2 on either side.

LML
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