Hi!

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

## Can a surface be formed by the intersection?

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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.

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.

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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.

### 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.

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

So the intesection of the paraboloid x

^{2}+y^{2}-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 x^{2}+y^{2}-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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### 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

http://en.wikipedia.org/wiki/Paraboloid

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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.

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