Quadrics in 4D and beyond

Higher-dimensional geometry (previously "Polyshapes").

Quadrics in 4D and beyond

Postby Marek14 » Wed Jul 20, 2005 9:35 am

My recent thoughts about classifying quadrics led me to make the assumption that each general class of quadric should be expressed with an equation of form sum(ai*xi^2)=b where each ai and b is -1, 0 or 1. Parabolic shapes have one of the summands linear instead of quadratic. (if there are multiple linear summands, they form a generic hyperplane and the shape can be rotated to make it parallel with one of the coordinates. To qualify as a quadric, there should be obviously at least one quadratic summand, I will relax this condition, however.

Naive enumeration would lead to 5^(n+1) possible equations for dimension n. First thing to do is to remove 3 (equations with 0 on the left) and cut the remaining number in half (since every equation can be multiplied by -1). Finally, we can disregard the order of coordinates on the left and only consider the combination of +xi^2, -xi^2, +xi, -xi, and zeros...

In 1D, we reach the following possible equations:

x=0
x=1
x=-1
x^2=0
x^2=1
x^2=-1

It's obvious that first four equations really describe the same thing - a single point. The fifth one describes a pair of points, while the sixth is an empty set of points. The important thing here is that these equations will appear later as cross-sections of 2D quadrics.

In 2D, we can copy these equations. In general, we get a cylinder whenever there are some summands with zero coefficient. 2D "cylinders" will be either a single line or a pair of parallel lines.

There are more interesting shapes equations possible:

x^2+y=0
x^2+y=1
x^2+y=-1

These are all parabolas. We can realise that the shape of figure is not affected by the right side, because we can rephrase the equation as
x^2=b-y
which has solutions for any b. Moreover, even the sign of "y" is meaningless. So we end with a single, most simple equation of parabola:

x^2-y=0 (there is minus because people prefer their parabolas to open in upwards direction)

If we will slide a line through this in y direction (y is fixed), we get the equation

x^2=y0

From this we can deduce that for negative y, we won't find any solutions, for y=0 exactly one, and for y positive we will always find two.

If we will move in x direction, keeping x fixed, then we get
y=x0^2
which has always a single solution.

If we have both x and y in quadratics, we will get the following shapes:

x^2+y^2=0 - point
x^2+y^2=1 - circle
x^2+y^2=-1 - empty set
x^2-y^2=0 - two intersecting lines
x^2-y^2=1 - hyperbole
x^2-y^2=-1 - also a hyperbole, it transforms in the first one by sign change and coordinate switch.

Slicing circle in any direction, we will find a single point splitting into two, which drift away before eventually coming back together and merge. Slicing a hyperbole depends on the direction - in both cardinal directions we will start with two points that move towards - but in one direction, they only come this close, then drift away, whereas in the other direction, they merge into one, for a time the 1D slice doesn't encounter any points of hyperbole, then a single point appears, once again splitting in two which drift away.

Two intersecting lines are the limit case - here, the two points merge, but immediatelly diverge again.

Going onwards to three dimensions.

First, the cylinders - by omitting one or two coordinates from the equation, allowing them to take any values at all, we arrive to one line, one plane, two parallel planes, parabolic cylinder, circular (or elliptic) cylinder, hyperbolic cylinder and two intersecting planes.

There are two kinds of paraboloids:
x^2+y^2-z=0 - elliptic paraboloid
x^2-y^2-z=0 - hyperbolic paraboloid

Slicing the elliptic one by z axis will find no solutions for z<0, a point for z=0, and circle for z>0. If we slice it in x or y direction, we find that all slices are parabolas, only differing in the position of the vertex which traces a parabolic path of its own.

Slicing hyperbolic paraboloid in z direction, we find a hyperbola whose two branches come closer until they merge in two straight lines at z=0. At this point, they "exchange partners" - for z<0, the branches go away in x direction, for z>0 it is in y direction.

Slicing from x or y we will find parabolas. All parabolic slices in one dimension will open either up or down. With this equation, slices for fixed x will open downwards, while those for fixed y will open upwards.

Apart from paraboloids, there is an ellipsoid
x^2+y^2+z^2=1, together with associated single point and empty set.

Then there is a cone
x^2+y^2-z^2=0
Sliced in x or y direction, it looks like hyperbole transforming in two straight lines and back again. From z direction, it's a circle shrinking into point and expanding again. The cone is a border between two kinds of hyperboloids...

One-part hyperboloid has equation x^2+y^2-z^2=1. Sliced from x (y slices are analogical), we start with hyperbole in yz plane opening in z direction. This will eventually transform in two straight lines and in hyperbole opening in y direction. After passing through x=0, the trend reverses, and the hyperbole becomes two lines and starts to open in z direction again.
Slicing in z direction will show you a circle shrinking to a minimum size, then expanding again.

Two-part hyperboloid has equation x^2+y^2-z^2=-1. Sliced from x, you get a hyperbole that always opens in the same direction. Sliced from z, you get a circle shrinking to a point, then, a while later, a point expanding into a circle. The figure is two-part since there is no solution for z=0, yet there are solutions for both positive and negative z.

So, how do we fare in 4D? Of course we get lots of cylinders based on lower-dimensional figures, but we are not interested in them that much.

First, the paraboloids: there are still only two.

x^2+y^2+z^2-w=0
x^2+y^2-z^2-w=0

We can call them "4-elliptic paraboloid and 4-hyperbolic paraboloid".

4-elliptic paraboloic, sliced from w, will look like a point suddenly appearing and expanding into an ellipsoid. Sliced from x, y, or z, it looks like a moving elliptic paraboloid in 3D.

4-hyperbolic paraboloid, sliced from w, starts its life as one-part hyperboloid for negative w, and passes through cone at w=0 to two-part hyperboloid for positive w. Sliced from x or y, you get a moving hyperbolic paraboloid, and sliced from z, you get a moving elliptic paraboloid.

After briefly mentioning 4-ellipsoid x^2+y^2+z^2+w^2=1, we get to three kinds of 4-hyperboloids, as well as two kinds of 4-cones.

The cones are
x^2+y^2+z^2-w^2=0
and
x^2+y^2-z^2-w^2=0

The first is the "light cone" in (3+1)-dimensional spacetime. Sliced from x, y, or z, we see a two-part hyperboloid transforming in a cone for coordinate=0, then separating again. Slicing through w, we see ellipsoid shrinking to a point, and expanding again.

The second kind of cone looks the same no matter which direction we slice it. It's a one-part hyperboloid transforming in a cone, then back again.

The three 4-hyperboloids are as follows:

x^2+y^2+z^2-w^2=1
x^2+y^2+z^2-w^2=-1
x^2+y^2-z^2-w^2=1

The first one, sliced through x, y, or z, is a two-part hyperboloid transforming via cone to one-part hyperboloid, then back. Sliced through w, it's ellipsoid shrinking to minimum size, then expanding again. This is 4D analogue of one-part hyperboloid.

The second hyperboloid, sliced through x, y, or z will be always two-part hyperboloid, and thus is a two-part figure. Sliced through w, it's ellipsoid shrinking into point, and a while later, a point expanding into ellipsoid. This is 4D analogue of two-part hyperboloid.

The third one is the most interesting. Sliced through x or y, you start with one-part hyperboloid which briefly passes through cone-bordered region of two-part hyperboloids. Slicing through z or w, however, gives a one-part hyperboloid that stays that way, with its central circle shrinking to minimum size, then expanding again.

Before I go, brief overview of 5D quadrics:

Paraboloids:
x^2+y^2+z^2+w^2-v=0
x^2+y^2+z^2-w^2-v=0
x^2+y^2-z^2-w^2-v=0

Ellipsoid:

x^2+y^2+z^2+w^2+v^2=1

Cones:
x^2+y^2+z^2+w^2-v^2=0
x^2+y^2+z^2-w^2-v^2=0

Hyperboloids:
x^2+y^2+z^2+w^2-v^2=1
x^2+y^2+z^2+w^2-v^2=-1
x^2+y^2+z^2-w^2-v^2=1
x^2+y^2+z^2-w^2-v^2=-1

Is this correct?
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Postby wendy » Wed Jul 20, 2005 11:17 am

I prolly would not know this one. not easily though. suppose i should, since the assorted conics are ultimately related to the class theorms.

In practice, i just imagine that there's one, and that we're seeing different projections of it.
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the dream we dream together is reality.

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Re: Quadrics in 4D and beyond

Postby quickfur » Tue Jan 31, 2006 8:25 pm

Hmm, I just noticed today that you have beaten me to classifying quadrics in n dimensions. :-) I didn't know you had already done the work before, and I'd just posted my independent discoveries yesterday.

Anyway, I can confirm that what you found is correct. There are 17 quadrics in 4D, 9 of which are cylinders of 3D quadrics, leaving 8 non-cylindrical quadrics: the hyper-ellipsoid, 5 hyperboloids (I count the cones together with these), and 2 paraboloids.

You are also right that in 5D there are the hyper-ellipsoid, 6 hyperboloids (including the 2 cones), and 3 paraboloids. There are also 17 non-degenerate cylinders, made from the 4D quadrics, making a total of 27 quadrics.

In general, in n dimensions, you get:
    - 1 hyper-ellipsoid
    - if n is odd:
      - 3(n-1)/2 hyperboloids (including cones)
      - (n-1)/2 + 1 paraboloids
    - if n is even:
      - 3(n-2)/2 + 2 hyperboloids (including cones)
      - n/2 paraboloids
      - and as many cylinders as there are quadrics in (n-1) dimensions.


Solving this recurrence yields the number of non-trivial quadrics to be Q(n) = n<sup>2</sup> + n - 1, or, if you discount degenerate quadrics, N(n) = n<sup>2</sup> + n - 3, for n > 1, and N(1) = 0.

("Trivial" here is defined as being empty or having only 1 point; "degenerate" is when the polynomial can be factored into a product of two hyperplanes.)
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