Enumeration of quadric manifold types in n dimensions

Higher-dimensional geometry (previously "Polyshapes").

Enumeration of quadric manifold types in n dimensions

Postby quickfur » Tue Jan 31, 2006 4:58 am

(I didn't realize phpBB imposed a length limit on the subject line... took a while to get one that fit. :?)

Over the past few days without an internet connection, I've been intensely interested in enumerating all possible types of quadric manifolds in 4D (i.e., those representable as a 2nd degree polynomial in 4 variables). As a result of a little independent research, I've come up with a scheme for enumerating all possible geometrically-distinct quadrics in n dimensions, and a formula that states how many non-trivial, non-degenerate distinct quadrics there are in n dimensions.

I'll state, without proof for now, the result: in n dimensions, the number of geometrically-distinct, non-trivial quadrics is:

n<sup>2</sup> + n - 1

where n >= 1.

(Trivial in this case means it either has no solutions, or contains only one point.)

Now, this count includes degenerate quadrics, which are products of 2 hyperplanes, corresponding with 2nd-degree polynomials that can be factored into a product of lines. The count without degenerate quadrics is:

0 (for n=1)
n<sup>2</sup> + n - 3 (for n > 1)

Plugging in n=2 and n=3 gives the familiar number of conic sections and quadric surfaces. Plugging in n=4 gives the number of quadric manifolds in 4D, which happens to be 17, broken down as follows:

1) The hyper-ellipsoid
2) Five hyperbolic manifolds (including the ellipsoidal cone)
3) Two parabolic manifolds
4) Nine non-degenerate cylinders of lower-dimensional quadrics. (I'm using "cylinder" here in the calculus sense; in wendy's terms it's the prism product.)

Among the cylinders are the spherinder (sphere-prism in wendy's terms), the cubinder (circle-square prism), and the conical cylinder (cone prism). The duocylinder is not representable by a single 2nd-degree polynomial in 4 variables.

One of the two parabolic manifolds I call the Supersaddle or Supercone (in analogy with how conic sections derive from the 3D cone): its intersections with different hyperplanes yield all 4 types of paraboloids and hyperboloids in 3D. The equation of the Supersaddle is:

x<sup>2</sup> + y<sup>2</sup> - z<sup>2</sup> - w = 0

Sadly, I could not find any 4D quadric that can give rise to all 9 3D quadrics. The Supersaddle is the closest candidate, but there is no ellipsoid in it. Interestingly enough, the 4D ellipsoidal cone can only yield hyperboloids of 2 sheets, 3D cones, and ellipsoidal paraboloids on intersection with hyperplanes. There's no way to get a hyperboloid of 1 sheet out of it, nor the 3D saddle.

Now, in 5 dimensions, there are:
- 1 hyper-ellipsoid (as always in every dimension)
- 6 hyperbolic quadrics
- 3 parabolic quadrics
- 17 non-degenerate cylinders (according to the number of quadrics in 4D)
giving a total of 27 quadrics.

If people are interested, I'll explain how I arrived at these numbers. :-)
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Postby thigle » Tue Jan 31, 2006 11:57 pm

i am interested, but prolly not skilled enough mathematically. but i have a question related to this:

...just as in the case of confocal conics in the plane, ...the confocal quadrics in the 3-space intersect each other at right angles, i.e. the tangent planes of the three surfaces passing through any given point in space are mutually perpendicular. (the points of the focal curves are exceptional: here 2 of the 3 planes are indeterminate.)
such systems of 3 mutually orthogonal families of surfaces - the systems of confocal quadrics being the outstanding exemple - occur in many mathematical and physical considerations. thus the analytical representation of these surfaces leads to "elliptic coordinates", which have proved very effective in the treatment of numerous problems,...
Hilbert & Cohn-Vossen_p22 in Geometry & Imagination, 1952


now my question is by dimensional analogy this: do any of the 17 quadric manifolds combine 'tetravalently' to form a coordinate system similar to the one mentioned in 3d by Hilbert ? how does one solve a system of 4 2nd degree polynomials in 4 variables ? which combinations work and which don't ?
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Postby quickfur » Wed Feb 01, 2006 12:12 am

thigle wrote:[...]
now my question is by dimensional analogy this: do any of the 17 quadric manifolds combine 'tetravalently' to form a coordinate system similar to the one mentioned in 3d by Hilbert ? how does one solve a system of 4 2nd degree polynomials in 4 variables ? which combinations work and which don't ?

Beats me. I haven't actually attempted to do any computation with these quadrics. I merely found a way to enumerate them by eliminating isomorphic quadrics from the set of all 2nd degree polynomials in 4 variables so that only the "representative" quadrics are left.

I'm still trying to understand the (+,+,-,-) hyperbolic quadric geometrically. It seems to be an interesting combination of cones and hyperboloids.
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