Mathematical Challenge (Concept only).

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zol
Posts: 40
Joined: Sun Mar 23, 2008 4:14 pm

Mathematical Challenge (Concept only).

Post by zol »

Hi everyone.

This is an idea that I had ages ago, but haven't had the time or inspiration to finish.

You know how to recursively quarter squares, right?
Now, if we move the centers to produce non-square quadrilaterals, we can get a non-affine warped grid/checkerboard effect.

I have the general outline of how to do it, but haven't thought of a good way to find a finite solution set.
My intuition is that there are solutions, but I have no proof.

Let me explain.

Within CF, we want a finite - preferably small - number of rules corresponding to different shapes and subdivisions.
Because of the way transforms are inherited, each rule need only deal with one canonical form, however distorted it appears on canvas.
The canonical forms of parent and child shapes with their corresponding subdivisions need to form a closed solution set to allow arbitrarily deep mutual recursion.

One possible canonical form is the union of a unit square and a triangle (a square with one side extended by some amount producing two non-right angles).
The inline transforms calling the main rules would use a combination of shear (sorry, 'skew') and scale to pass quarters in canonical form (plus any required re-alignment).

That's not a very explicit description, sorry, and maybe I forgot some details.
(But if your imagination doesn't fill in the blanks, you probably wouldn't have much luck solving it anyway.)

Any takers? 8)

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Nom
Posts: 30
Joined: Tue Jan 30, 2007 6:45 am

Post by Nom »

Must be possible to some extent, if I correctly understand what you mean.

Have you encountered the classic integral approximation of circle that uses cardinal 3 as a basis, progressively moving along two spatial axises ?

I think that acquiring the maths for such like algorithms will be quite heavy work ?

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zol
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Joined: Sun Mar 23, 2008 4:14 pm

Post by zol »

I can't think of which approximation you mean, although 2d approximation is familiar enough and might be necessary to get numerical solutions for whatever (in)equalities are found.

The first problem, I think, is getting some more constraints on the solution so that it's easier to characterise.

I suggested a single type of canonical form (possibly too restrictive, with two adjacent right angles) to this end.

But it doesn't help much with the means of discovery:
-Combine an arbitrary shape with an arbitrary 'centre'.
-Normalise each quarter in up to 8 variants (4 rotation x 2 reflection).
-Check whether each has a variant already in the set.
-For any quarter with no variants in the set add one of its variants (which one?) and apply the same procedure using another (arbitrary) 'centre' for the new shape.
-If closure doesn't happen soon enough, start again with different shape and/or partition.

The main problem is knowing whether the set is finite (or can at least fake it within the level of precision).
A better characterisation could dispose of some 'arbitraries', and possibly lead to an algebraic solution.

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Nom
Posts: 30
Joined: Tue Jan 30, 2007 6:45 am

Post by Nom »

I was (rather vaguely, pardon me) referring to Bresenham's circle algorithm. But it's really out of scope with context free, involving a conditional. Generally I'd say that the complexity of possible patterns is *very* severely limited by exclusion of conditionals of any kind (including backreference to stored values). But your case is yours to prove :)

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