Not fractal but funny; general torus knots DE routine

[cKleinhuis]

i tried the d.IFS.torusIFS, i just got a single torus, and could change the position, hybrids where copies, and the box parameters gave me the rectangular torus??

[DarkBeam]

 

Supershape is much easier to do, because it's an implicit surface. Well it should be at least for an incomplete implement

Boxy are ON/OFF settings set a nozero to enable

For getting a knot use KnotsIFS.m3f

Parametric surfaces must be inverted to give a DE, that's hard to do

Not all knots are possible with my routine. Example of what you cannot get http://it.wikipedia.org/wiki/File:Figure_8_knot.png 

[cKleinhuis]

where is the "delete formula" button ?!?!

[DarkBeam]

  Why delete? You can hide it - right click on the list to show a popup menu

[DarkBeam]

Hey! I think it is the figure eight knot;


Mandelbulb3Dv17{
V.....S....O/.../....2.....fqMzHsm8A.b468pAn0C3EfvlKe7t5HxvZ8LzOGylIz8j/4aX0MR0k
................................OaNaNaNadz1........A./..................y.2.....
................/M.0/....6kl....8.....E2.....QzIXLua2PrD/..........m/dkpXm1....U
z.....kD12../..........wz..................................u1....y1...sD...../..
.w1...sDECayXi7lqx1..........WjJy0vma8ejv8jrnNkXCr9YXdzcPGgRzEEnhwuma8ej2oQ9jigd
WuXyKt9g9PecyYmQF6ua2PrD......Y0..............kD.2....sD..kz0...................
.............oAnAr1...sD....z.MRg4.6qlO..P5f/.qRg4..slO.UW5f/.ISg4....../....k1.
..................kz.wzzz1.U..6.P....M4...EB....W....k1....F....8/...I1.....Sl42
...U.qFG9yzb2zzzRYoWzz7lz16.mc..zXCc.El18XGQeGyDjvIRhrVAkz1..........2.28.kFrA0.
FWb96aAIVzvh1se7Umvxz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................2CcN/UvNP6.eeWCNq0.yRii.EJJUk1f..XR
S1.mx3CcN/UvNP6.QsLsUa3.ibhV..bTV1OK.sSq40.ly3CcN/UvNP6.MwLsUa3.ibhV.kqTV1OK.sSq
40.kz3CcN/UvNP6...EsUa3.eeWCNq0.IJ36wk8.wyLsUa3.................................
E....6....E.....I....g....kGix4RnZYFH/kPrJaQ..........................U1C....6U.
06.................4./.........E........................................kz1.....
...wz........U.E........kz1........3.LaNaNaNaN.k................................
.....................2.....3....7....EbQVtqQaxaQhZYFH/..........................
....4MU/06U..................................................U.E........E.2.....
...0.1..........................................................................
................................}

How cutie

[Syntopia]

Thanks, DarkBeam - it is a great system.

Here is a GLSL version:

Code:

void rotate(inout vec2 v, float angle) {
  v = vec2(cos(angle)*v.x+sin(angle)*v.y,-sin(angle)*v.x+cos(angle)*v.y);
}

float DE(vec3 p) {
float mobius = (a+b/polyfoldOrder) * atan(p.y,p.x);
p.x = length(p.xy)-R1;
rotate(p.xz,mobius);
float m = polyfoldOrder/(2.*PI);
float angle = floor(.5+m*(PI/2.-atan(p.x,p.z)))/m;
rotate(p.xz,angle);
p.x =p.x - R3;
return length(p.xz)-R2;
}

...now, what about a DE for these Coxeter polytopes?:
http://www.math.cmu.edu/~fho/jenn/index.html

[cKleinhuis]

ehrm, how to combine those DEs with standard fractals

[subblue]

Yet again very impressive stuff DarkBeam! Syntopia, your renderer is looking very tasty

[DarkBeam]

syntopia you are kidding right?
Those polytopes look damnedly complicated to me.
Anyway do you know what is the 4d to 3sphere projection algorithm? Jesse was looking for it . thanks

[Jesse]

Anyway do you know what is the 4d to 3sphere projection algorithm? Jesse was looking for it . thanks

Really, can't remember?

ehrm, how to combine those DEs with standard fractals

DE combo... (maybe in future: use the coords of the other fractal after n iterations or something)

[subblue]

Using Syntopia's GLSL snippet in Knighty's Pseudo Kleinian example

[DarkBeam]

Omg... Jesse and Subblue!
Techniques used?

Edit for Jesse; see this about projections; http://www.dimensions-math.org/Dim_CH3_E.htm

Even better this; with formulas!!! http://teamikaria.com/hddb/forum/viewtopic.php?f=5&t=61

Code:

pt3.x = pt4.x-((pt4.w-ppdist)*(pt4.x-obspt.x)/(pt4.w-obspt.w))
pt3.y = pt4.y-((pt4.w-ppdist)*(pt4.y-obspt.y)/(pt4.w-obspt.w))
pt3.z = pt4.z-((pt4.w-ppdist)*(pt4.z-obspt.z)/(pt4.w-obspt.w))

ppdist = the w value of the picture realm.
Make it less than the smallest point on your object (my tesseract's points are 1 or -1 in each directionm, and so I set this value at -2..)
Then the observation point (mine is set at 0,0,0,-4)
What these formulas are doing is to find out where a line crosses the picture realm when drawn from a point on the object
 to the observer's position. If you think about it, this is why we see things the way we do.
Everything we see is gathered to a point (our eye). :-) To further reduce it to 2D (for display on a monitor...)

pt2.x = pt3.x-((pt3.z-ppdist)*(pt3.x-obspt.x)/(pt3.z-obspt.z))
pt2.y = pt3.y-((pt3.z-ppdist)*(pt3.y-obspt.y)/(pt3.z-obspt.z))

[cKleinhuis]

any example m3f files ?!

so, those de function cant be used to be incorporated in the fractals ?
they just can be used in a layered manner ?!

.... btw. what a about a layering .... *duck away....*

Anyway do you know what is the 4d to 3sphere projection algorithm? Jesse was looking for it . thanks

Really, can't remember?

ehrm, how to combine those DEs with standard fractals

DE combo... (maybe in future: use the coords of the other fractal after n iterations or something)

[cKleinhuis]

who was asking the place is dying somehow ?!?!?!? !

[DarkBeam]

Oh, it was dying before the superhero rise

Uhm, btw, I was looking again at the theory and the parametric plotter to verify the figure8 thingy. Yes, my program cannot do it. But why afterall?

Uploaded at ImageFra.me

So in synthesis;

Some knots look identical seen in x-y projection but are different, because z spins twice as fast than x,y

And how in the world I can make it spin faster without touching other components?

I tried to apply the identity sin(4t)=2 sin(2t) cos(2t) = 2 sin(2t) sqrt(1- sin(2t)^2 )

But if I replace z*z with 4*z*z*(1-z*z) very weird things happen, (because DE is supposed to be positive outside the knot) and in section the torus does not show plain circles anymore. Ideas?

(enclosed image of what I get roughly applying the z mangling)