by quickfur » Fri Nov 18, 2011 12:27 am
Hmm, you're right, mrrl, even in 2D you do have the parallel lines quadric, which is not generated by cone sections.
Anyway, I started digging in my old notes, and found a little analysis for classifying all quadrics in n dimensions that I did many years ago. Basically i reduce any n-dimensional quadric to a standard form via an affine transformation. Under such a transformation, we can separate all variables of a quadratic polynomial in n variables to get a sum of polynomials in each variable. Furthermore, we can eliminate all except maybe 1 linear term (e.g., x^2+y+z=K can be reduced to x^2+y=K'). If there is a linear term, then it's a "parabolic" manifold; otherwise, it's some combination of spherical and/or hyperbolic.
The rest of the classification is then based on the "signature" of the polynomial, basically the configuration of positive or negative terms for each squared variable. For example, x^2+y^2-z^2 has signature (++-) and x^2-y^2-z^2 has signature (+--). The order of signs in the signature doesn't matter, so we can always write the +'s first. There's an interesting relationship between quadrics if we consider the operation of negating the signature. For example, the negation of (++-) is (--+), which is the same as (+--). So the hyperboloid of 1 sheet is, in some sense, "conjugate" or "dual" of the hyperboloid of 2 sheets.
A cylinder is basically a lower-dimensional quadric embedded in a higher dimension (since the unspecified variable gives it the freedom of extrusion).
The combination of signature and type (parabolic/non-parabolic) defines a class of quadrics, and may be homogenous or not (constant term is 1 or 0). Self-dual signatures give rise to two members, homogenous and non-homogenous; non-self-dual signatures, if considered as a pair with their dual, gives 3 members: homogenous, non-homogenous, and dual non-homogenous.
There are also 3 types of degenerates: product of two coincident hyperplanes (which gives a perfect square (ax+y)^2=0); product of two parallel hyperplanes, and product of two non-parallel hyperplanes.
By counting the number of possibilities in each dimension, I obtained the number of quadrics in each dimension as P(n) = n^2+n-1. If we exclude the degenerates, then the number of quadrics is Q(n) = 0 for n=1, and Q(n) = n^2+n-3 for n>1. Q(1)=0 because 1D only has the degenerate quadric (x/a)^2=1. (I'm not 100% confident about Q(n) for n>1, though, I think i may have made a mistake in solving the recurrence.)
Anyway, the 4D quadrics are classified thus:
Category 1a: (++++): the 3-ellipsoid: only 1 member.
Category 1b: three types: (+++-), (++--), and (+---).
(+++-) is dual to (+---), and gives 3 types of quadrics: w^2+x^2+y^2-z^2=1, w^2-x^2-y^2-z^2=1 (dual to each other), and the homogenous case: w^2+x^2+y^2-z^2=0 (shape of the light cone).
(++--) is self-dual; giving us two members: w^2+x^2-y^2-z^2=0 and w^2+x^2-y^2-z^2=1.
So there are 5 members in category 1b.
Category 2: parabolics, with signature (+++) and (++-), giving us w^2+x^2+y^2-z = 0, and w^2+x^2-y^2-z=0.
Category 3: cylinders of 3D quadrics: there are 11 of these, 2 of which are degenerate.
So in total, there are 17 kinds of quadrics in 4D, or 19 if you include the two degenerates.