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I'm sure we've all seen the image below that illustrates the creation of the four conic sections. Although I've seen this multiple times throughout my education, I find it odd that the following case has never been discussed: If we take the vertical plane that forms a hyperbola and tilt it just a few degrees in a way that still crosses both sides of the cone, shouldn't we get an asymmetric hyperbola? Being an astronautical engineer, I've studied how (symmetric) hyperbolas model highly eccentric orbits of celestial objects. What physical processes can be modelled by asymmetric hyperbolas (assuming they exist)?

Conics Image

LeonardBlunderbuss
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No, we get perfectly symmetric hyperbolas for any plane intersecting a double cone on both of its parts.

viplov_jain
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  • That can't be true. Look at the plane creating the hyperbola above. If you rotate the plane at the center of the cone, you could get to a point where the plane just touches the cone at the bottom and creates a larger parabola at the top. Anywhere in between we would have two different-sized parabolas. – LeonardBlunderbuss Feb 19 '14 at 17:38
  • It's impossible that any non-vertical plane would create symmetric parabolas on both sides of the cone, unless the plane crosses the center of the cone. – LeonardBlunderbuss Feb 19 '14 at 17:39
  • The only case a plane just touches a cone is when it makes a pair of straight lines. – viplov_jain Feb 19 '14 at 18:15
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The comments by the OP indicate that he may be confused about the nature of the branches of a hyperbola. A branch of a hyperbola is never a parabola. One way of seeing this is to notice that a branch of a hyperbola has a pair of transverse asymptotic lines, whereas a parabola does not have asymptotic lines at all.

When one tilts the vertical plane it may look as if one is creating an asymmetric hyperbola, but this is an illusion. The illusion is possibly due to the fact that one tends to think of the center of the hyperbola is being "at the same level" as the vertex of the cone. What actually happens is that the center of the hyperbola travels away from this level as the plane is tilted more and more.

Mikhail Katz
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  • Ah, I see now! When the plane is tilted, you must extend the length of the cone on one side to see the symmetry. I don't know why that wasn't obvious to me before. Thanks so much! – LeonardBlunderbuss Feb 19 '14 at 18:02