Is a cone the degenerate of a 4 dimensional hyperbola?

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SUMMARY

A cone is indeed considered a degenerate form of a 4-dimensional hyperbola. The mathematical representation x² + y² - z² = C illustrates that when C equals zero, the hyperboloid transitions into a cone, confirming the relationship between these geometric shapes. The discussion emphasizes that degenerates typically exist one dimension lower than their originating forms, supporting the conclusion that a cone is a valid degenerate of a 4-dimensional hyperbola. Additionally, the complexity of visualizing higher dimensions is acknowledged, particularly regarding the limitations of representing cones in 4D space.

PREREQUISITES
  • Understanding of hyperboloids and their equations, specifically x² + y² - z² = C.
  • Familiarity with the concept of degeneracy in geometry.
  • Knowledge of dimensionality in geometric shapes.
  • Basic skills in higher-dimensional visualization techniques.
NEXT STEPS
  • Research the properties of hyperboloids and their degenerates in higher dimensions.
  • Explore the mathematical implications of C values in hyperboloid equations.
  • Study higher-dimensional geometry and visualization methods.
  • Learn about the relationship between cones and other geometric forms in various dimensions.
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying higher-dimensional spaces will benefit from this discussion, particularly those interested in the properties and relationships of geometric shapes.

JonDrew
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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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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.
 
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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