Intersecting ellipsoids/quadrics

Discussion of shapes with curves and holes in various dimensions.

Intersecting ellipsoids/quadrics

Postby mr_e_man » Wed Jan 04, 2023 12:43 am

(This came up while I was studying 4D rotational dynamics. But that's a different topic.)

I have a 3D subset of 6D space, given as the intersection of two spherical cylinders (I'm not sure if this is the right terminology) and an ellipsoid:

x₁² + x₂² + x₃² = c₁²,
x₄² + x₅² + x₆² = c₂²,

(x₁ - x₄)²/(a₁² + a₄²) + (x₂ - x₅)²/(a₂² + a₄²) + (x₃ - x₆)²/(a₃² + a₄²)
+ (x₁ + x₄)²/(a₂² + a₃²) + (x₂ + x₅)²/(a₃² + a₁²) + (x₃ + x₆)²/(a₁² + a₂²) = 1.

What I want to know is whether this 3D subset splits into several pieces. Is it always connected? or is it always disconnected? or does it depend on the shape parameters c₁,c₂,a₁,a₂,a₃,a₄ ?

(I think we shouldn't count degenerate cases, where slightly changing the parameters changes the connectivity.)


For comparison, consider a 1D subset of 3D space given as the intersection of a circular cylinder and an ellipsoid:

x₁² + x₂² = c₁²,

x₁²/a₁² + x₂²/a₂² + x₃²/a₃² = 1.

(WolframAlpha plot.) This always splits into 2 separate curves, except in degenerate cases where the curves meet at 4-way intersections, or where the cylinder is tangent to the ellipsoid everywhere on its equator.
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Re: Intersecting ellipsoids/quadrics

Postby mr_e_man » Thu Jan 05, 2023 9:07 pm

It's not always connected.
The first two equations (x₁²+x₂²+x₃²=c₁² and x₄²+x₅²+x₆²=c₂²) imply that the x's are comparable to the c's in magnitude.
If a₃ and a₄ are very large compared to the other parameters, then the third equation takes the form (small stuff) + (x₃+x₆)²/(a₁²+a₂²) = 1. (This means that the ellipsoid is very wide, approximately a pair of parallel (hyper)planes.) So the 3D subset splits into two pieces; one piece has x₃+x₆ ≈ +√(a₁²+a₂²), and the other piece has x₃+x₆ ≈ -√(a₁²+a₂²).

Is it ever connected?

The subset as a whole is centrally symmetric; x can be replaced with -x.
Even if it's never connected, can each piece be centrally symmetric?
In other words, if the c's and a's are specially chosen (but not too precisely, to avoid degeneracy), can every point be connected to its opposite point?
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