Hypercone

In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation

It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named spherical cone because its intersections with hyperplanes perpendicular to the w-axis are spheres. 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

with vertex at the origin and expansion speed s.

An oblique spherical hypercone could then be described by the function

where ${\displaystyle (v_{x},v_{y},v_{z})}$ 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.

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