# Hypercone

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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.

## Parametric formEdit

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.

## Geometrical interpretationEdit

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 nonzero *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 1/3π*r*^{4}, 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.

### ConstructionEdit

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 interpretationEdit

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:

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 alsoEdit

## ReferencesEdit

**^**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.