Curve Filling a Rectangle

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jewish_scientist
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Curve Filling a Rectangle

Postby jewish_scientist » Fri Apr 20, 2018 2:56 pm UTC

The Hilbert curve completely fills a square. Can a modified version be used to fill a rectangle? My instinct says yes, but I wanted to check anyway.
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Re: Curve Filling a Rectangle

Postby doogly » Fri Apr 20, 2018 3:29 pm UTC

Sure, the Hilbert curve is defined with an x(s) and y(s), and if you take two bump functions and feed those in, you can get what you want.
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Re: Curve Filling a Rectangle

Postby Soupspoon » Fri Apr 20, 2018 3:43 pm UTC

Given the lines are either horizontal or vertical, and you steadily fill the gaps between horizontals by vertical fractures and the gaps between the verticals by horizontal fractures, I'd say that taking a non-unitXunit ratio box and progressively filling it with similarly ratioed higher-order curves (applied as a transform in the same orientation as the box, i.e. complimentary ratios as you recurve around the corner of the bigger curve before it) would hit total horizontal filling by infinite widthless vertical line-segments at the same time as vertical filling by the similar stack of horizontal ones.

Or, by another way of looking at it, if ∞ = 4∞ (which it does, arguably, from various standard usages of aleph-null) then ∞*(1/2)=∞*(2/1), so a 1:2 rectangle gets filled just as much at the absolute limit of space-filling in both axes.

But I can also imagine counter-interpretations. Hilbert curves might not work, but Peany ones would?

PTW, then.

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Re: Curve Filling a Rectangle

Postby madaco » Mon Apr 23, 2018 8:53 pm UTC

The square and the rectangle are homeomorphic.

Take the obvious homeomorphism between the square and the rectangle. Compose this with the curve. The result should be a curve that fills the rectangle.

Right?
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Re: Curve Filling a Rectangle

Postby Eebster the Great » Tue Apr 24, 2018 3:28 am UTC

Couldn't you just substitute, say, x/2 for x? I'm missing the reason why you have to actually do anything at all.

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Re: Curve Filling a Rectangle

Postby Xanthir » Tue Apr 24, 2018 6:28 pm UTC

aka what madako said, yeah. It's a trivial mapping.
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Re: Curve Filling a Rectangle

Postby MartianInvader » Wed Sep 26, 2018 2:41 am UTC

Just wanted to add, the beauty of a space-filling curve is that it's purely topological - which means you can compose it with any continuous function (well, any surjective continuous function) and you still have a space-filling curve.

So you can start with the Hilbert curve and "stretch" it to fill a rectangle, a triangle, a circle, a star, or pretty much any other connected 2d region.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Curve Filling a Rectangle

Postby jewish_scientist » Fri Sep 28, 2018 5:42 pm UTC

I did not know that. Thanks for the information.
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Re: Curve Filling a Rectangle

Postby Eebster the Great » Fri Sep 28, 2018 9:08 pm UTC

By "surjective continuous function," do you mean in one variable (i.e. ℝ→ℝ or [0,1]→ℝ)? If you mean ℝ→ℝ2, then by surjectivity you already have a space-filling curve.

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Re: Curve Filling a Rectangle

Postby Xanthir » Fri Sep 28, 2018 10:44 pm UTC

Presumably ℝ²→ℝ²; that's the only thing that typechecks when composed with the space-filling curve, which maps [0,1]→ℝ²
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Re: Curve Filling a Rectangle

Postby Eebster the Great » Sat Sep 29, 2018 12:38 am UTC

Yeah it wasn't clear to me in which order he was composing the functions, but in this case, again, that's just what surjective means.

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Re: Curve Filling a Rectangle

Postby MartianInvader » Sat Sep 29, 2018 8:22 pm UTC

Eebster the Great wrote:Yeah it wasn't clear to me in which order he was composing the functions, but in this case, again, that's just what surjective means.

I meant from the square to a region of R2. And being surjective isn't enough, as I said it also needs to be continuous. Otherwise, you don't have a space-filling curve, just a space-filling... function, I guess.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Curve Filling a Rectangle

Postby Eebster the Great » Sun Sep 30, 2018 7:30 am UTC

Fair enough. Essentially, you can transform any curve filling any (simply-connected open) subset of R2 to one filling any other by just composing it with any continuous function from the one onto the other. That such a function always exists follows from the Riemann Mapping Theorem, and for most practical subsets, it's really easy to find such a continuous function.

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Re: Curve Filling a Rectangle

Postby MartianInvader » Sun Sep 30, 2018 4:58 pm UTC

You can even do a little better than what the Riemann mapping theorem gets you, since you only need a surjective mapping and not a full-on homeomorphism. You can wrap the square around a donut shape (annulus), for example (you don't need simple connectedness, just path connectedness I believe).
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Curve Filling a Rectangle

Postby Eebster the Great » Mon Oct 01, 2018 12:33 am UTC

Right, the Riemann Mapping Theorem actually gives you a biholomorphic function. Are biholomorphic functions the isomorphisms of complex analysis?

What is a sufficient condition for the existence of a continuous surjection that is stricter than the hypothesis of the Riemann Mapping Theorem?

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Re: Curve Filling a Rectangle

Postby MartianInvader » Mon Oct 01, 2018 9:58 pm UTC

Eebster the Great wrote:What is a sufficient condition for the existence of a continuous surjection that is stricter than the hypothesis of the Riemann Mapping Theorem?

We're being a little fast and loose with our conditions anyway, since the Riemann Mapping theorem is about open sets, but anything covered by a path is going to be compact (and therefore closed). Anyways, I'm pretty sure that any compact path-connected set can be surjected onto by the square, although the only way I can think to prove it is to essetially construct a space-filling curve directly.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Soupspoon
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Re: Curve Filling a Rectangle

Postby Soupspoon » Mon Oct 01, 2018 11:54 pm UTC

Have you tried logarithms?

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Re: Curve Filling a Rectangle

Postby MartianInvader » Tue Oct 02, 2018 4:18 pm UTC

Oh my goodness, of course! Logarithms! I don't know why I didn't see it before! Soupspoon, you're brilliant! And definitely someone who knows what they're talking about and not at all faking it.

I'm going to have to move you way up in the rankings in my latest project - ranking people from best to worst.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Curve Filling a Rectangle

Postby Soupspoon » Tue Oct 02, 2018 4:40 pm UTC

Qapla'!

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gd1
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Re: Curve Filling a Rectangle

Postby gd1 » Tue Oct 02, 2018 11:42 pm UTC

Soupspoon wrote:Qapla'!


Please hold your Qapla's until after the lecture.
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Re: Curve Filling a Rectangle

Postby mashnut » Tue Oct 16, 2018 12:52 am UTC

gd1 wrote:
Soupspoon wrote:Qapla'!


Please hold your Qapla's until after the lecture.


That's how I got kicked out of the last lecture though :(

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Re: Curve Filling a Rectangle

Postby DavCrav » Sat Oct 27, 2018 7:24 pm UTC

Soupspoon wrote:Have you tried logarithms?


To be fair, taking the log of one of the variables would indeed map a square onto a rectangle.


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