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April 1

Other extension of cone to 'hypercone'

The article Hypercone extends the concept of a cone z^2=x^2+y^2 to z^2=w^2+x^2+y^2. However, I'm interested in the other extension, w^2+z^2=x^2+y^2. Does it have a name?

While I can imagine the hypercone sections fairly clearly by doing t^2=x^2+y^2+z^2 (t being time giving shrinking then expanding sphere), I'm having problems with imagining the sections of the other extension. Any suggestions?Naraht (talk) 14:12, 1 April 2014 (UTC)[reply]

You should be able to visualize one set of parallel sections as a hyperboloid of revolution, scaling as with the sphere. —Quondum 14:28, 1 April 2014 (UTC)[reply]

The "other" has parallel sections corresponding to hyperboloids of revolution of one sheet with t=0 corresponding to the infinite double cone and the abs(t) increasing "expanding the hole" in the middle of the one sheet.

OTOH, the intersections for the hypercone if the variable with is treated like t is on the other side (z^2=x^2+y^2+t^2), then you get the hyperboloids of revolution of two sheets "inside" the cones...

I think a standard name would be ultrahyperbolic cone (or ultrahyperbolic null cone). More generally, you can consider the null cone associated to any nondegerenerate quadratic form. Over the reals, these are characterized by their signature. The (space like) cross sections are always (affine) quadrics. The null cross sections are actually just affine spaces. Sławomir Biały (talk) 19:39, 1 April 2014 (UTC)[reply]