Undergrad Converting 2 COD (x,y) into 1 Hilbert curve COD?

[greswd]

COD stands for co-ordinate.

As the title says, you have two co-ordinates of a point, x and y, on a unit square.

What's the formula for converting these two co-ordinates into a single Hilbert curve co-ordinate?

Which represents the percentile along the length of the Hilbert Curve that point is on.

 

[greswd]

Inspired by this video

 

[BvU]

What's the formula for converting

Would this be converting 2D to 1D ?

I'm inlcined to think along lines like: write x and y as real numbers. Create a z as follows: first digit is first digit of x, second digit is first digit of y -- third digit is second of x, fourth second of y, etc. etc.

Ever hear of the Hilbert hotel ?

 

[WWGD]

Isn't this just the arc length parametrization of the Hilbert curve? Not that it is simple but it seems like a useful way of framing it.

 

[BvU]

Ah, I completely misread this thing -- totally off.
Thanks @WWGD

So @greswd , you want a function of $\ x, y\$ and $n$ that returns a fraction for a Hilbert curve of order $n$ starting at the lower left ?

Start with $n = 1$, then $2$ etc and see if you can find a pattern

 

[BvU]

Found some time to watch the video -- cute !
I understand you want the limit for $n\uparrow \infty$ (the ones with finite $n$ are pseudo Hilbert curves). Advice is the same.

Found one value already: $f(1/2, 1/2) = 1/2$

 

[BvU]

Susspect the time 4:00 -- 5:00 min in the video is important: suppose you work with base 4 numbers and work out the diagonal flip of first and last quadrant ...then make it recursive ..

Just a wake-up thought ...

Intriguing !

 

[greswd]

Isn't this just the arc length parametrization of the Hilbert curve? Not that it is simple but it seems like a useful way of framing it.

If arc length parametrization is what the video is doing, then yes

 

[greswd]

Ah, I completely misread this thing -- totally off.
Thanks @WWGD

So @greswd , you want a function of $\ x, y\$ and $n$ that returns a fraction for a Hilbert curve of order $n$ starting at the lower left ?

Start with $n = 1$, then $2$ etc and see if you can find a pattern

thanks, do you know the function for x and y in terms of the fraction and n?

 

[BvU]

No idea yet -- first trying to get a good and complete problem statement in my chaotic mind .

Anyway, PF isn't about doing the work for you, it's about helping you to do the work
(however much we'd like to grab it and do it ourselves )

If I ever have the time, I think I'd start reading this
The heatmap here is nice too

But you already googled those, right ?

 

[greswd]

No idea yet -- first trying to get a good and complete problem statement in my chaotic mind .

Anyway, PF isn't about doing the work for you, it's about helping you to do the work
(however much we'd like to grab it and do it ourselves )

If I ever have the time, I think I'd start reading this
The heatmap here is nice too

But you already googled those, right ?

yeah I have haha. The problem is "infinitely" difficult.

Consider the ratio along the length of the HC, L.
From L, you can get a value for X and a value for Y.

So, you get a function for X in terms of L and the same for Y. Next, the inverse functions, L in terms of X and L in terms of Y.
There are an infinite number of inverse functions.

You get two sets of infinite equations, one from X and one from Y.

There can only be one specific value of L which can be the solution to the equations in both infinite sets. And it is the solution to only one equation in each set.

This is way beyond me lol. I just want to know the formula for L in terms of X and Y.