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Is a cone the degenerate of a 4 dimensional hyperbola?

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Jan30-13, 10:48 PM
P: 83
Is a cone a the degenerate of a 4 dimensional hyperbola?

I only ask because I think it is and I am not sure. I am trying to get better at higher dimensional visualizations.

My analogy being that a point is the degenerate of a 3 dimensional cone. With that logic wouldn't that make a cone the degenerate of a 4 dimensional hyperbola?
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Jan31-13, 08:52 PM
P: 233
Sort of, though not 4 dimensions, but 3.

x^2+y^2-z^2=C is a hyperboloid of two sheets if C<0, one sheet if C>0, and a cone when C=0.
Jan31-13, 09:05 PM
P: 83
Aren't degenerates usually at least one dimension less than what they degenerate from? and If not could it still be the degenerate of a 4 dimensional hyperbola?

Because I don't think a cone can exist in 4 dimensions, it would be too many axes going through a single point, right?

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