Curve Filling a Rectangle

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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?

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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.

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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.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

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