Spherical Cone - Parametric Form

Parametric Form

A right spherical hypercone can be described by the function

with vertex at the origin and expansion speed s.

An oblique spherical hypercone could then be described by the function

where is the 3-velocity of the center of the expanding sphere. An example of such a cone would be an expanding sound wave as seen from the point of view of a moving reference frame: e.g. the sound wave of a jet aircraft as seen from the jet's own reference frame.

Note that the 3D-surfaces above enclose 4D-hypervolumes, which are the 4-cones proper.