Hypercone

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A hypercone is a higher-dimensional generalization of a cone. Specifically, a four-dimensional right spherical hypercone can be thought of as a sphere which expands with time, starting its expansion from a single point source, such that the center of the expanding sphere remains fixed. An oblique spherical hypercone would be a sphere which expands with time, again starting its expansion from a point source, but such that the center of the expanding sphere moves with a uniform velocity.

A right spherical hypercone can be described by the function

$\vec \sigma (\phi, \theta, t) = (t s \cos \theta \cos \phi, t s \cos \theta \sin \phi, t s \sin \theta, t)$

with vertex at the origin and expansion speed s.

An oblique spherical hypercone could then be described by the function

$\vec \sigma (\phi, \theta, t) = (v_x t + t s \cos \theta \cos \phi, v_y t + t s \cos \theta \sin \phi, v_z t + t s \sin \theta, t)$

where $(v x, v y, v y)$ 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.