A hypercone consists of a large donut-shaped balloon that supports a cone-shaped sheet of heat-resistant fabric thirty to forty meters in diameter, with the capsule located at the point of the cone.
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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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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.
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While I can imagine the hypercone sections fairly clearly by doing t ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 ( t being time giving shrinking then expanding sphere ), I'm having problems with imagining the sections of the other extension.
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The hypercone is intended to supplement other deceleration mechanisms, bridging a gap in capability between conventional heat shields ( which are useful for taking a spacecraft from orbital velocity down to several times the speed of sound ) and conventional parachutes or landing rockets ( which are only useful below the speed of sound ).
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:: OTOH, the intersections for the hypercone if the variable with is treated like t is on the other side ( z ^ 2 = x ^ 2 + y ^ 2 + t ^ 2 ), then you get the hyperboloids of revolution of two sheets " inside " the cones . ..
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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.
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This diagram is plotted in special relativistic x and t coordinates, with the future at the top and the past at the bottom, and light travels along 45?diagonal lines . ( To get the full four-dimensional version of the diagram, rotate it in the third dimension around a vertical axis through the center to get a cone shape, then rotate that through the fourth dimension to get a hypercone .)
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