Is a cone the degenerate of a 4 dimensional hyperbola?

JonDrew

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?

Tinyboss

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.

JonDrew

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?

Similar Threads for: Is a cone the degenerate of a 4 dimensional hyperbola?
Thread	Forum	Replies
Hyperbola and cone	General Math	0
dimensional analysis and frustum of a cone	Introductory Physics Homework	4
Non-degenerate and degenerate perturbation theory	Quantum Physics	13
parameterization of hyperbola intersecting cone	Calculus & Beyond Homework	5
non-degenerate Poisson bracket and even-dimensional manifold	Differential Geometry	1

Tinyboss	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.

JonDrew	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?