Alain Martel <kua### [at] videotronca> wrote:
> Do you mean the repeating gradient ?

> larger than 1 get their integer part zeroed.
>
> Then, it's mapped to a 0..0.5 range.
>
> It could look something like this :
> colour_map{
>  [0 hsv2rgb(<Hvalue, 1, 0>)]
>  [1 hsv2rgb(<Hvalue, 1, 0.5>)]
>    }

I mean, looking at the bottom-left corner, or the concentric pattern in the cut
at the top of the singularities, I see it as a superposition of several
surfaces. Also, the shape of the singularities seems to have slight steps, at
the border of what I believe to be each layer. But the more I look at it the
more I feel like I'm just completely confused by the repeating colour pattern
and the shape of the isosurface... which makes this image even more intriguing
:-)

Pascal

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On 22/12/2021 06:51, Tor Olav Kristensen wrote:
> Hi
> 
> Since this post:
> 
> From: kurtz le pirate
> Subject: How to ...
> Date: 2021-11-22 11:23:08
> http://news.povray.org/povray.general/thread/%3C619bc3ec%241%40news.povray.org%3E/
> 
> - I've been working on some some macros that create functions for calculating
> with complex numbers.
> 
> And yesterday I made some functions that can be used for HSV-coloring of
> pigments.
> 
> The isosurface in the attached image shows the magnitude (or modulus) of this
> function:
> 
>   Fn(Z) = 1/(Z^5 - 2)^2
> 
> - as the height above a complex plane:. I found that function here:
>
https://matlabarticlesworld.blogspot.com/2020/01/what-is-coolest-thing-you-can-do-with.html
> 
> The colors are chosen so that the hue follows the phase (or argument) of the
> function, while the lightness goes from 0.0 to 0.5 in intervals along the height
> axis. The saturation is 100% everywhere.

Very good job !

The basic operators on the complexes is much more elaborate than mine. I
use simple macros. This makes the definitions a little more difficult to
work out. For example, for f(z) = z + 1/z, I have to write
complexAdd(cc,complexInverse(cc)).

The really interesting part is the use of isosurfaces. Good job. I just
used colored triangles which give me a {} mesh. The coloring model is
also very clever because the hue and luminosity depend on the value of
the function.

I will see how to add the same coloring scheme as you.

Here, one sample of f(z) = (-z^3 + iz^2 + 1) / (z - 1 + i)^2.




-- 
Kurtz le pirate
Compagnie de la Banquise

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Attachments:
Download 'complex3dmesh_02_fn2.jpg' (48 KB)

Preview of image 'complex3dmesh_02_fn2.jpg'

"BayashiPascal" <bai### [at] gmailcom> wrote:
>...
> Very nice.
>
> The "multi-layered" aspect of the result is intriguing me. Does it come from a
> property of the function you've choosen, or from the way you choose to visualise
> it ?

Thank you Pascal

The layered appearance is just a result from the coloring.
Here's how I did it:

#declare H_Fn =
    function(re, im) {
        degrees(mod(2*pi + ArgumentFn(re, im), 2*pi))
    }
;
#declare S = 1.0;
#declare A = 0.6; // 0 < A < 1
#declare L_Fn =
    function(re, im) {
        mod(10*(1 - pow(A, MagnitudeFn(re, im))), 1)/2
    }
;
isosurface {
    function { y - MagnitudeFn(x, z) }

    ...

    FunctionsPigmentRGB(
        function { Rd_Fn(H_Fn(x, z), S, L_Fn(x, z)) },
        function { Gn_Fn(H_Fn(x, z), S, L_Fn(x, z)) },
        function { Bu_Fn(H_Fn(x, z), S, L_Fn(x, z)) }
    )
}

In the attached image L_Fn() was changed to this:

#declare A = 0.2; // 0 < A < 1
#declare L_Fn =
    function(re, im) {
        (1 - pow(A, MagnitudeFn(re, im)))
    }
;

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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Attachments:
Download 'fivepoles_isosurface_otherlightness.png' (202 KB)

Preview of image 'fivepoles_isosurface_otherlightness.png'

Alain Martel <kua### [at] videotronca> wrote:

> > "Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
> >>...
> >> And yesterday I made some functions that can be used for HSV-coloring of
> >> pigments.
> >>...
> >
> > The "multi-layered" aspect of the result is intriguing me. Does it come from a
> > property of the function you've choosen, or from the way you choose to visualise
> > it ?
> >...
> Do you mean the repeating gradient ?

> larger than 1 get their integer part zeroed.
>
> Then, it's mapped to a 0..0.5 range.
>
> It could look something like this :
> colour_map{
>  [0 hsv2rgb(<Hvalue, 1, 0>)]
>  [1 hsv2rgb(<Hvalue, 1, 0.5>)]
>    }

Sorry Alain. I made a mistake and wrote "HSV-coloring" above. I meant to write
"HSL-coloring". (I haven't yet looked at how HSV formulas works.)

Also see my answer to Pascal in my previous post.

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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"Kenneth" <kdw### [at] gmailcom> wrote:
> "Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
>
> >
> > The colors are chosen so that the hue follows the phase (or argument) of the
> > function, while the lightness goes from 0.0 to 0.5 in intervals along the height
> > axis. The saturation is 100% everywhere.
> >
>
> Those beautiful color gradations remind me of old-style blown-glass/metallic
> Christmas tree ornaments; it even appears as if they have blurred reflections.
> That's an amazing result. Nice!

Thank you Kenneth !

The nice color gradients resulting from the formulas was a surprise to me.

I don't think that I have seen such old style blown glass ornaments.
Are they like this ?
https://www.nordichouse.co.uk/vintage-bordeaux-glass-bauble-p-4022.html

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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"Bald Eagle" <cre### [at] netscapenet> wrote:
>...
> Very nice.

Thank you Bill !


> I was consulting the stuff that Paul Nylander wrote.  I'm assuming yours are
> similar.

I'm not very familier wit Paul Nylander's work. Those macros seem like a good
start for a library for complex calculations. But the Pow() macro could need
some work to allow for the exponent to also be a complex number.

I did not create macros to do the calculations, but arrays of functions and
macros that assemble functions into new functions. For each complex operator
there's two functions; one for calculating the real part and one for calculating
the imaginary part.


>...
> I made these two to just keep track
>
> #macro Argument (Re, Im)
>  atan2 (Re, Im)
> #end
>
> #macro Modulus (Re, Im)
>  sqrt (pow (Re, 2) + pow (Im, 2))
> #end

I like your Modulus() macro better than the Abs() macro, because it does not
rely on the underlying implementation of how the complex numbers are
represented. I think that as few as possible of the macros should depend on the
underlying implementations. Btw.: Why have you chosen to have a different
atan2() call in your Argument() macro than in the Arg() macro ?


> I worked those out from the macros in colors.inc.  A little challenging at first
> to turn that whole thing into a function.  ;)

Yes, that's a bit of a struggle.


>...
> This is looking great!  I'm sure there are a lot of other interesting complex
> surfaces to be explored.

I've started on a Github repository for my library. It's here:
https://github.com/t-o-k/POV-Ray-complex-functions

Please note that this is a work in progress, so some features hasn't been added
yet and much of it may change.


> I'm also wondering how hard it would be to use mod()
> to have an infinite array of those "black hole vortices" on a plane - in either
> a rectangular or an alternating/hexagonal arrangement...

That's an interesting idea: to have a mod() operator that can handle complex
values. But I don't know how to implement that...

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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kurtz le pirate <kur### [at] gmailcom> wrote:
>...
> Very good job !
>
> The basic operators on the complexes is much more elaborate than mine. I
> use simple macros. This makes the definitions a little more difficult to
> work out. For example, for f(z) = z + 1/z, I have to write
> complexAdd(cc,complexInverse(cc)).

Yes, that's like a prefix notation, which is natural for such a macro library.


> The really interesting part is the use of isosurfaces. Good job. I just
> used colored triangles which give me a {} mesh. The coloring model is
> also very clever because the hue and luminosity depend on the value of
> the function.

Thank you. I'm glad that you like this Kurtz.

It wanted my library to also work with isosufaces and pigments, so I had to use
functions to do all the calculations.

I read about domain coloring with HSL-colors here:

https://en.wikipedia.org/wiki/Domain_coloring


> I will see how to add the same coloring scheme as you.

You could have a peek at my implementation here:

https://github.com/t-o-k/POV-Ray-complex-functions


> Here, one sample of f(z) = (-z^3 + iz^2 + 1) / (z - 1 + i)^2.

Here's how I have to enter that function in order to process it with my current
macros. It's a kind of postfix notation/implementation.

#declare No = 16;

#declare PartTypes = array[No];
#declare Arguments = array[No];

#declare PartTypes[0] = "Z";
#declare Arguments[0] = ZFn();

#declare PartTypes[1] = "Const";
#declare Arguments[1] = RealConstFn(+3.0);

#declare PartTypes[2] = "Pow";
#declare Arguments[2] = Arg2Fn(0, 1);

#declare PartTypes[3] = "Neg";
#declare Arguments[3] = Arg1Fn(2);

#declare PartTypes[4] = "Z";
#declare Arguments[4] = ZFn();

#declare PartTypes[5] = "Const";
#declare Arguments[5] = ImagConstFn(+1.0);

#declare PartTypes[6] = "Sqr";
#declare Arguments[6] = Arg1Fn(4);

#declare PartTypes[7] = "Mul";
#declare Arguments[7] = Arg2Fn(5, 6);

#declare PartTypes[8] = "Add";
#declare Arguments[8] = Arg2Fn(3, 7);

#declare PartTypes[9] = "Const";
#declare Arguments[9] = RealConstFn(+1.0);

#declare PartTypes[10] = "Add";
#declare Arguments[10] = Arg2Fn(8, 9);

#declare PartTypes[11] = "Z";
#declare Arguments[11] = ZFn();

#declare PartTypes[12] = "Const";
#declare Arguments[12] = ComplexConstFn(-1.0, +1.0);

#declare PartTypes[13] = "Add";
#declare Arguments[13] = Arg2Fn(11, 12);

#declare PartTypes[14] = "Sqr";
#declare Arguments[14] = Arg1Fn(13);

#declare PartTypes[15] = "Div";
#declare Arguments[15] = Arg2Fn(10, 14);

I'm a bit worried though, because my renderings of that function is quite
different from mine.

What is your rendering showing ?
The magnitude, the real part or the imaginary part ? - or something else ?

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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"Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
>...
> #declare PartTypes[4] = "Z";
> #declare Arguments[4] = ZFn();
>
> #declare PartTypes[5] = "Const";
> #declare Arguments[5] = ImagConstFn(+1.0);
>
> #declare PartTypes[6] = "Sqr";
> #declare Arguments[6] = Arg1Fn(4);
>
> #declare PartTypes[7] = "Mul";
> #declare Arguments[7] = Arg2Fn(5, 6);
>

Oops.
I should have written those assignments like this:

#declare PartTypes[4] = "Const";
#declare Arguments[4] = ImagConstFn(+1.0);

#declare PartTypes[5] = "Z";
#declare Arguments[5] = ZFn();

#declare PartTypes[6] = "Sqr";
#declare Arguments[6] = Arg1Fn(5);

#declare PartTypes[7] = "Mul";
#declare Arguments[7] = Arg2Fn(4, 6);

- But the resulting functions would yield the same values.

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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"Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
> Thank you Pascal
>
> The layered appearance is just a result from the coloring.

Thank you ! Now my confusion is entirely cleared up. :-)

Pascal

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"Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmailcom> wrote:
> "Kenneth" <kdw### [at] gmailcom> wrote:
> >
> > Those beautiful color gradations remind me of old-style blown-glass/metallic
> > Christmas tree ornaments...
>
> I don't think that I have seen such old style blown glass ornaments.

These three examples come closest to what I remember seeing, when I was a kid
visiting my grandmother for the holidays...

https://www.pinterest.com/pin/210824826294535957/

https://www.pinterest.com/pin/645281452835747592/

https://www.pinterest.com/pin/550002173251947194/

.... but your colors are even better and more saturated. And you have managed to
capture that blurred metallic 'sheen'. It's probably an optical illusion, but
the effect is quite magical.

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