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Is a cone the degenerate of a 4 dimensional hyperbola? 
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#1
Jan3013, 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? 


#2
Jan3113, 08:52 PM

P: 233

Sort of, though not 4 dimensions, but 3.
x^2+y^2z^2=C is a hyperboloid of two sheets if C<0, one sheet if C>0, and a cone when C=0. 


#3
Jan3113, 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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