Another way of formulating the 2-D conic sections is that they are all points such that the ratio of the distance from a fixed point
F (the focus) and the distance from a fixed line
D (the directrix) (not through
F) is constant.
Of course, moving this all up a dimension, we are looking for all points such that the ratio of the distance from a fixed point
F and the distance from a fixed realm
D (not containing
F) is constant.
WLOG, assume that
D is the realm
w = 0 and that
F is the point
( 0, 0, 0, p ) and the said ratio is
e > 0. Then, the general conic section is all points
( x, y, z, w ) satisfying:
e<sup>2</sup>w<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + ( w - p )<sup>2</sup>
e = √2/2, p = 1
In the special case where
e = 1, this is the parabolic shape:
w = ( x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + p<sup>2</sup> ) / p
e = 1, p = 1
In the case where
0 < e < 1, this is a non-spherical ellipsoid. In the case where
e > 1, this is a hyperbolic solid.
e = √6/2, p = 1
Defined this way, the conic sections do not get any more interesting as one goes up in dimensions. There are only two free parameters, the focal point's distance from the plane and the ratio. Now, things would definitely get more interesting if we let the directrix
D be any flat space of dimension less than
n and we let
F be any flat space of dimension less than that of
D. These wouldn't be
conic any longer, but they'd get more interesting.