Hypercone

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This article is about the geometric concept. For the spacecraft re-entry mechanism, see Hypercone (spacecraft).
Stereographic projection of a spherical cone's generatrices (red), parallels (green) and hypermeridians (blue). Due to conformal property of Stereographic Projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone. The hypermeridians generates a set of concentric spheres.
Stereographic projection of a spherical cone's generatrices (red), parallels (green) and hypermeridians (blue). Due to conformal property of Stereographic Projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone. The hypermeridians generates a set of concentric spheres.

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

x2 + y2 + z2 − w2 = 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.

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[edit] 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 (vx,vy,vy) 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.

[edit] 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 πr4 / 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.

A 3D cone is made by either stacking circles of successively smaller sizes from base to top (a point), or by wrapping one edge of a triangle around a sphere and connecting the other two edges to each other, or you can use the stacking method and wrap a triangle around the outside of the shape.

All that you need to do to make a hypercone is bump the same components up a dimension. make the circles spheres, the line a circle, and the triangle a tetrahedron. so start with the circle (in the 3D cone it is the line along which the circles are stacked) and stack spheres that decrease in size as you go in either direction along the circle out from the base sphere (in the 3D cone this sphere would be the base circle). After the stacking is finished wrap a tetrahedron around it. The tetrahedron has a fat end (any given face) and a skinny end (the vertex opposite that face). The circle with spheres along it also has a skinny point where the spheres become so small they become a line, and it has a fat end at the base. Put the stacked spheres and circle they are stacked along inside the tetrahedron matching each of their skinny ends and fat ends to be at the same end of the shape. then to finish making the hypercone, wrap the tetrahedron (into the fourth dimension) around the stacked spheres. This is an illustration of it folding.

[edit] Temporal interpretation

If the w-coordinate of the equation of the spherical cone is interpreted as time, then it is the shape of the light cone in special relativity. In this case, the equation is usually written as:

x2 + y2 + z2 − t2 = 0.

The upper nappe is then the future light cone and the lower nappe is the past light cone.

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