Can you take two hyperbolas an make an ellipse?

Question

I guess I know in some way this is a silly question, but technically if I take a cylinder I can cut diagonally and get two hyperbolas (with two half of the cylinder) and if I change the angle of the cut I can make an ellipse (from the slices), thus would that mean that 2 hyperbolas make an ellipse?

Answer 1

I am trying to imagine what you are trying to imagine ..

The curve of intersection is an ellipse intrinsically, but not a hyperbola, no matter how you place it..

I thought that you thought...by taking two cut half-cylinders and placing them opposite each other as shown you see a figure similar to a hyperbola and so you can always get back the ellipse.

Similar is not same. The way intersection curve bends is quite different.our

EDIT 2:

Ah! now seem to be getting some more insight into your perceptions..through the Projections!

You can in fact project a hyperbolic arc on a plane to get it to a parabola, an ellipse or a circle by placing a light beam or torch at the vertex of a cone on a rigid arc of a curve and adjusting the inclination of the plane of its shadow...

Projns of Conic arcs

Answer 2

No, two hyperbolas don't form an ellipse.

Both hyperbolas and ellipses are conic section. A conic section is uniquely defined by five points on the curve (as long as no four of them are collinear). So if you have a piece of a hyperbola, you can pick five points on it, build the conic through these, and will get the full hyperbola. You can't get anything else if you start with five points on a hyperbola.

If you take a second hyperbola, and just take a piece from one and a piece from the other, that doesn't help. There is only one conic to fit each piece of hyperbola, and that's the hyperbola itself. An ellipse won't fit to a piece of hyperbola.