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. {\displaystyle 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 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 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
σ → ( ϕ , θ , t ) = ( t s cos θ cos ϕ , t s cos θ sin ϕ , t s sin θ , t ) {\displaystyle {\vec {\sigma }}(\phi ,\theta ,t)=(ts\cos \theta \cos \phi ,ts\cos \theta \sin \phi ,ts\sin \theta ,t)}
with vertex at the origin and expansion speed s. A right spherical hypercone with radius r and height h can be described by the function
σ → ( ϕ , θ , t ) = ( t cos ϕ sin θ , t sin ϕ sin θ , t cos θ , h r t ) {\displaystyle {\vec {\sigma }}(\phi ,\theta ,t)=\left(t\cos \phi \sin \theta ,t\sin \phi \sin \theta ,t\cos \theta ,{\frac {h}{r}}t\right)}
An oblique spherical hypercone could then be described by the function
σ → ( ϕ , θ , t ) = ( v x t + t s cos θ cos ϕ , v y t + t s cos θ sin ϕ , v z t + t s sin θ , t ) {\displaystyle {\vec {\sigma }}(\phi ,\theta ,t)=(v_{x}t+ts\cos \theta \cos \phi ,v_{y}t+ts\cos \theta \sin \phi ,v_{z}t+ts\sin \theta ,t)}
where ( v x , v y , v z ) {\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.
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 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 1/3πr4, 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 axis of symmetry. 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 3D cone standing upright in the 4th direction as it rotates freely about any of its planes of symmetry.
Measurements
Hypervolume The hypervolume of a four-dimensional pyramid and cone is
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