Curve Filling a Rectangle
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Curve Filling a Rectangle
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
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
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 nonunitXunit ratio box and progressively filling it with similarly ratioed higherorder 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 linesegments 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 alephnull) then ∞*(1/2)=∞*(2/1), so a 1:2 rectangle gets filled just as much at the absolute limit of spacefilling in both axes.
But I can also imagine counterinterpretations. Hilbert curves might not work, but Peany ones would?
PTW, then.
(Ninja says it more succinctly than me.)
Or, by another way of looking at it, if ∞ = 4∞ (which it does, arguably, from various standard usages of alephnull) then ∞*(1/2)=∞*(2/1), so a 1:2 rectangle gets filled just as much at the absolute limit of spacefilling in both axes.
But I can also imagine counterinterpretations. Hilbert curves might not work, but Peany ones would?
PTW, then.
(Ninja says it more succinctly than me.)
Re: Curve Filling a Rectangle
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?
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
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
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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