## Can a surface be formed by the intersection?

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mathmari
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### Can a surface be formed by the intersection?

Hi!

I have question.. Can a surface be formed by the intersection of a paraboloid and a plane?

cyanyoshi
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### Re: Can a surface be formed by the intersection?

Are you asking whether the intersection of a paraboloid (2-dimensional) and a plane (also 2-dimensional) can be another 2-dimensional surface? I guess that depends whether you consider a plane as a special case of a paraboloid. If not, then I am inclined to say no.

Non-degenerate paraboloids have non-zero curvature, unlike planes. Suppose that you could find a 2-dimensional region that a plane and a paraboloid have in common. You could then find the intrinsic curvature of this intersection. If this curvature is zero, then it could not be a finite 2-D chunk of the paraboloid, but if this curvature is non-zero, then it can't be part of the plane. Therefore by contradiction, the region of intersection must not be a surface.

Carmeister
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### Re: Can a surface be formed by the intersection?

This might be slightly unrelated, but it made me wonder... what is the intersection of a paraboloid and a plane in 3d euclidean space? Depending on the angle it could be a parabola or a circle... would something in between would give you something like an ellipse, but I don't think it would be an ellipse since that would be the intersection of a plane with a cone... but I could be wrong there.

Moole
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### Re: Can a surface be formed by the intersection?

Carmeister wrote:This might be slightly unrelated, but it made me wonder... what is the intersection of a paraboloid and a plane in 3d euclidean space? Depending on the angle it could be a parabola or a circle... would something in between would give you something like an ellipse, but I don't think it would be an ellipse since that would be the intersection of a plane with a cone... but I could be wrong there.

If you were to write out the equation for an arbitrary paraboloid, it would be a degree two polynomial in its three coordinates; in the coordinates of the plane, it would still be a degree two polynomial, meaning it would have to be a conic section (since the conic sections are exactly the curves of algebraic degree 2). Therefore it would, in fact, be an ellipse.
Mathematical hangover (n.): The feeling one gets in the morning when they realize that that short, elementary proof of the Riemann hypothesis that they came up with at midnight the night before is, in fact, nonsense.

mathmari
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### Re: Can a surface be formed by the intersection?

So the intesection of the paraboloid x2+y2-z=0 and the plane z=2 is a circle? So can it not be a surface?

jestingrabbit
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### Re: Can a surface be formed by the intersection?

mathmari wrote:So the intesection of the paraboloid x2+y2-z=0 and the plane z=2 is a circle? So can it not be a surface?

Not unless you are so lax in you definition of paraboloid that you include planes as a kind of paraboloid.

But I should warn that whilst there are elliptic paraboloids, which when intersected with a plane yield an ellipse or a parabola, there are also hyperbolic paraboloids. These can yield a parabola, a line, two intersecting lines, or a hyperboloid when intersected with a plane, but not an ellipse.
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z4lis
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### Re: Can a surface be formed by the intersection?

Pictures and equations of the two types of paraboloids jestingrabbit mentions:

http://en.wikipedia.org/wiki/Paraboloid
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Rhombic
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### Re: Can a surface be formed by the intersection?

Carmeister wrote:This might be slightly unrelated, but it made me wonder... what is the intersection of a paraboloid and a plane in 3d euclidean space? Depending on the angle it could be a parabola or a circle... would something in between would give you something like an ellipse, but I don't think it would be an ellipse since that would be the intersection of a plane with a cone... but I could be wrong there.

It would be egg-shaped!!
An ellipse with a larger diameter and a smaller one.