Can two ellipsoids meet in a pair of ellipses intersecting in four points?
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On the plane two ellipses can intersect in exactly four different points.
In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^2+(y/4)^2+z^2=1$ and $(x/4)^2+y^2+z^2=1$.
However, it seems impossible for two ellipsoids to intersect in a pair of ellipses intersecting in exactly four different points.
Intuitively it seems this is due to the fact that in such case the ellipses would lie on the same affine plane in space. What is the rigorous way to prove this?
linear-algebra geometry geometric-topology topological-vector-spaces
asked Aug 17 at 2:21
John
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On the plane two ellipses can intersect in exactly four different points.
In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^2+(y/4)^2+z^2=1$ and $(x/4)^2+y^2+z^2=1$.
However, it seems impossible for two ellipsoids to intersect in a pair of ellipses intersecting in exactly four different points.
Intuitively it seems this is due to the fact that in such case the ellipses would lie on the same affine plane in space. What is the rigorous way to prove this?
linear-algebra geometry geometric-topology topological-vector-spaces
asked Aug 17 at 2:21
John
1,503923
Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
On the plane two ellipses can intersect in exactly four different points.
In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^2+(y/4)^2+z^2=1$ and $(x/4)^2+y^2+z^2=1$.
However, it seems impossible for two ellipsoids to intersect in a pair of ellipses intersecting in exactly four different points.
Intuitively it seems this is due to the fact that in such case the ellipses would lie on the same affine plane in space. What is the rigorous way to prove this?
linear-algebra geometry geometric-topology topological-vector-spaces
asked Aug 17 at 2:21
John
1,503923
On the plane two ellipses can intersect in exactly four different points.
In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^2+(y/4)^2+z^2=1$ and $(x/4)^2+y^2+z^2=1$.
However, it seems impossible for two ellipsoids to intersect in a pair of ellipses intersecting in exactly four different points.
Intuitively it seems this is due to the fact that in such case the ellipses would lie on the same affine plane in space. What is the rigorous way to prove this?
linear-algebra geometry geometric-topology topological-vector-spaces
asked Aug 17 at 2:21
John
1,503923
asked Aug 17 at 2:21
John
1,503923
asked Aug 17 at 2:21
John
1,503923
asked Aug 17 at 2:21
John
1,503923
1,503923
Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56
add a comment |Â
Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56
Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56
add a comment |Â
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Did you read the descriptions of these tags at all? What does this have to do topological vector spaces, geometric topology, linear algebra?
– saulspatz
Aug 17 at 4:19
Why do you say it does not have anything to do with those areas?
– John
Aug 17 at 5:52
Because it doesn't. This is just a problem in analytic geometry, so far as I can see.
– saulspatz
Aug 17 at 7:01
Based on what I know about those areas I opine differently.
– John
Aug 17 at 13:56