Hypercone

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

\( x^2 + y^2 + z^2 - w^2 = 0 \)

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.

Parametric form

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 \( (v_x, v_y, v_z) \) 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.
Geometrical interpretation

The spherical cone consists of two unbounded nappes, which meet at the origin and are the analogues of the nappes of the 3-dimensional conical surface. The upper nappe corresponds with the half with positive w-coordinates, and the lower nappe corresponds with the half with negative w-coordinates.

If it is restricted between the hyperplanes w=0 and w=r for some non-zero r, then it may be closed by a 3-ball of radius r, centered at (0,0,0,r), so that it bounds a finite 4-dimensional volume. This volume is given by the formula\( \pi r^4/3 \), and is the 4-dimensional equivalent of the solid cone. The ball may be thought of as the 'lid' at the base of the 4-dimensional cone's nappe, and the origin becomes its 'apex'.

This shape may be projected into 3-dimensional space in various ways. If projected onto the XYZ hyperplane, its image is a ball. If projected onto the XYW, XZW, or YZW hyperplanes, its image is a solid cone. If projected onto an oblique hyperplane, its image is either an ellipsoid or a solid cone with an ellipsoidal base (resembling an ice cream cone). These images are the analogues of the possible images of the solid cone projected to 2 dimensions.

Construction

The (half) hypercone may be constructed in a manner analogous to the construction of a 3D cone. A 3D cone may be thought of as the result of stacking progressively smaller discs on top of each other until they taper to a point. Alternatively, a 3D cone may be regarded as the volume swept out by an upright isosceles triangle as it rotates about its base.

A 4D hypercone may be constructed analogously: by stacking progressively smaller balls on top of each other in the 4th direction until they taper to a point, or taking the hypervolume swept out by a tetrahedron standing upright in the 4th direction as it rotates freely about its base in the 3D hyperplane on which it rests.
Temporal interpretation
Main article: Minkowski space

If the w-coordinate of the equation of the spherical cone is interpreted as the distance ct, where t is coordinate time and c is the speed of light (a constant), then it is the shape of the light cone in special relativity. In this case, the equation is usually written as:

\( x^2 + y^2 + z^2 - (ct)^2 = 0, \)

which is also the equation for spherical wave fronts of light.[1] The upper nappe is then the future light cone and the lower nappe is the past light cone.[2]
See also

3-sphere
Hypercube
Hyperplane
Hypersurface
Manifold
Light cone

References

A. Halpern (1988). 3000 Solved Problems in Physics. Schaum Series. Mc Graw Hill. p. 689. ISBN 978-0-07-025734-4.
R.G. Lerner, G.L. Trigg (1991). Encyclopedia of Physics (2nd ed.). VHC publishers. p. 1054. ISBN 0-89573-752-3.

Mathematics Encyclopedia