On the following website definitions are given for the three non-degenerate conic sections. They are defined differently based on the ways the cutting plane is parallel to the generators. While I understand the cases of ellipse and parabola, how can the hyperbola be generated when the cutting plane is at the same time parallel to two generators of the cone? Can anyone explain this?
Here's a picture of what I wrote in my comment above. A plane $\alpha$ cuts a double cone along a hyperbola (blue). A plane parallel to $\alpha$ and passing through the vertex cuts the cone along two generatrices (orange).