Ellipse function in space?

Question

x^2+y^2=r^2 where r=radius if add z axis and span pararell to z say -a~+a then shape is like "can". if put straw on to "can" perpendicular to y_axis,shift this straw direction of x_axis and rotate "straw" to z_axis in y_z plane then intersect ellipse(?) ?

Answer 1

Two equations:

x²+y²=r²

y²+z²=q²

=> x²-z²=r²-q²

therefore from the side (x-z-plane), the intersect is a hyperbola. From the front (y-z-plane) the intersect is a circle ( the "straw" definition ). The precise shape will depend on the realation of the size of the straw to the can, is r=q, r<q or r>q.

The intersect is not contained in  a 2-dimensional-plane (no ellipse).