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Is a cone the degenerate of a 4 dimensional hyperbola?

  1. Jan 30, 2013 #1
    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. jcsd
  3. Jan 31, 2013 #2
    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.
     
  4. Jan 31, 2013 #3
    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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