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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?

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Michael Munta
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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).

enter image description here

Intelligenti pauca
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  • I'm not sure you know what I am asking. :) I have edited my question with a picture, look at c. They say that the plane which creates a hyperbola is parallel to 2 generatrices. So I am not looking for a case when the plane passes through the vertex. I would like to know which are those two generatrices that the plane is parallel with. – Michael Munta Apr 14 '20 at 20:52
  • The plane of the hyperbola IS IN FACT PARALLEL TO THE ORANGE GENERATRICES. Are you not convinced of that? – Intelligenti pauca Apr 14 '20 at 21:23
  • I rotated the picture, in case the different orientation confounded you. – Intelligenti pauca Apr 14 '20 at 21:26
  • I see it more clearly now, thanks. – Michael Munta Apr 15 '20 at 06:09