POV-Ray : Newsgroups : povray.advanced-users : Infinite poly-torus Server Time
17 Apr 2021 06:50:33 EDT (-0400)
  Infinite poly-torus (Message 1 to 3 of 3)  
From: Bald Eagle
Subject: Infinite poly-torus
Date: 28 Feb 2021 08:05:01
Message: <web.603b9438d59be3661f9dae300@news.povray.org>
No idea what to actually call it, but I had the idea to expand the Triple Torus

https://en.wikipedia.org/wiki/Genus_g_surface#Genus_3

into an infinite plane of smoothly blobbed tori.

That would tie in with the hexagonal tilings we've played with in the past, as
they'd have the same centers, and then I guess there would have to be some sort
of mod() operator to extend the function out to infinity.

[Can such a function be modified with a warp to make it wrap around another
torus or a cylinder...?]

That got me playing with some simple grid functions I found on ShaderToy, and we
really don't have any native grid pigment patterns in POV-Ray, which is odd.

I have something worked out to complement jr's recent re-working of Friedrich
Lohmueller's gridded scene-layout stuff, but it's in layered texture
{pigment{function{}}} form.

Seems like it works on all axes, with square and/or rectangular layout.
Maybe I should try skew just to see what, if anything breaks.

Maybe next will be triangular, hexagonal, ...


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From: Le Forgeron
Subject: Re: Infinite poly-torus
Date: 1 Mar 2021 07:10:39
Message: <603cd9bf@news.povray.org>
Le 28/02/2021 à 14:01, Bald Eagle a écrit :
> we
> really don't have any native grid pigment patterns in POV-Ray, which is odd.

There is the pavement pattern, or did I misunderstand the request ?

pigment { pavement number_of_sides 4 number_of_tiles 1 pattern 1
 ...

}


plane { y, 0 texture { pavement number_of_sides 6 number_of_tiles 4
pattern 2 texture_map{[0.5 T_Stone19]
[1 T_Stone20]
} scale 1/12 rotate y*-30 } }


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From: Bald Eagle
Subject: Re: Infinite poly-torus
Date: 1 Mar 2021 14:00:00
Message: <web.603d39589aa36b281f9dae300@news.povray.org>
Le_Forgeron <jgr### [at] freefr> wrote:
> Le 28/02/2021 à 14:01, Bald Eagle a écrit :
> > we
> > really don't have any native grid pigment patterns in POV-Ray, which is odd.
>
> There is the pavement pattern, or did I misunderstand the request ?

Nope - you just caught me sleeping.
"gradient" is a simple pattern, and I would have expected an earlier
implementation of a simple grid

The pavement patterns that you coded are - complex and bewildering in their
algorithmic intricacy.   I just hadn't made the connection between the
ultra-complex pavement tilings and a simple set of perpendicular lines.

Thanks.  :)


With regard to the torus shape, I'm maybe halfway to simply getting alternating
centerpoints for the tori.

As Mike Williams has very helpfully provided:

"It's possible to use a single isosurface to produce multiple copies of a shape
by using the mod operator in a substitution.
Using mod(x,2) will cause the shape to be repeated in the x direction every 2
units. If your original isosurface created a shape centered at the origin, then
there's a problem with the way that it chooses which bits to repeat. The left
half of the object gets repeated in one direction and the right half of the
object gets repeated in the other direction. To fix this, we'd like to
substitute mod(x,2)+1 where x is negative and mod(x,2)-1 where x is positive. To
fix both directions at once, for a symmetrical object, we can use abs(x) like
this mod(abs(x),2)-1.

To change the length of the repeat unit we can do this mod(abs(x),Step)-Step/2"



But what I (think I) need is one column at one step, and the next column at
Step+Step/2, then Step, Step+Step/2, etc....


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