Title: End of Greatness
Post by: Alef on December 30, 2012, 05:32:59 PM
Experimenting with neutron star gravitational waves formula by kram.
http://www.fractalforums.com/mandelbrot-and-julia-set/swirlbrot/ (http://www.fractalforums.com/mandelbrot-and-julia-set/swirlbrot/)
Made of KIFS, triplex and hudge amount of swirling chaotic mass of humanure. Something like this Vector=scale(gravitswirl(Vector))+C When I first saw this, it made me laught. But 2D zooms suggest some neat shapes.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_3.jpg)
Main problem with this is not enought thick solids, I believe, regular hole-solid structure is not becouse of discontiniuaties, but becouse of too thin shells sometimes unnoticed by brute force raytracer optimisation (it don't checks every voxel). And maybe so far too chaotic.
Iterate:
radius = cabs(z); (true modulus)
angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(zx+1i*zz);
zx= cos(frequency*radius+ spin*angx) *radius;
zy= cos(frequency*radius+ spin*angy) *radius;
zz= cos(frequency*radius+ spin*angz) *radius;
zx=zx*scale+Cx;
zy=zy*scale+Cy;
zz=zz*scale+Cz;
z= quaternion ( zx , zy, zz, 0);
With some tweaking I got this.
With maxiter=5, frequency=1, spin=1, scale=2.15, bailout =12.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_8.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/EOG_5.jpg)
And with maxiter=4, frequency=3.5, spin=3, scale=1.7, bailout =8.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_9.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/EOG_6.jpg)
If radius= (zx^8+zy^8)^0.125 then it creates more pillow like shapes.
http://www.fractalforums.com/mandelbrot-and-julia-set/swirlbrot/ (http://www.fractalforums.com/mandelbrot-and-julia-set/swirlbrot/)
Made of KIFS, triplex and hudge amount of swirling chaotic mass of humanure. Something like this Vector=scale(gravitswirl(Vector))+C When I first saw this, it made me laught. But 2D zooms suggest some neat shapes.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_3.jpg)
Main problem with this is not enought thick solids, I believe, regular hole-solid structure is not becouse of discontiniuaties, but becouse of too thin shells sometimes unnoticed by brute force raytracer optimisation (it don't checks every voxel). And maybe so far too chaotic.
Iterate:
radius = cabs(z); (true modulus)
angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(zx+1i*zz);
zx= cos(frequency*radius+ spin*angx) *radius;
zy= cos(frequency*radius+ spin*angy) *radius;
zz= cos(frequency*radius+ spin*angz) *radius;
zx=zx*scale+Cx;
zy=zy*scale+Cy;
zz=zz*scale+Cz;
z= quaternion ( zx , zy, zz, 0);
With some tweaking I got this.
With maxiter=5, frequency=1, spin=1, scale=2.15, bailout =12.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_8.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/EOG_5.jpg)
And with maxiter=4, frequency=3.5, spin=3, scale=1.7, bailout =8.
(http://www.ljplus.ru/img4/a/s/asdam/EOG_9.jpg)
(http://www.ljplus.ru/img4/a/s/asdam/EOG_6.jpg)
If radius= (zx^8+zy^8)^0.125 then it creates more pillow like shapes.
Title: Re: End of Greatness
Post by: Alef on January 02, 2013, 09:04:22 AM
Well, now I changed gravitational waves formula so that y is calculated by sin (aka mathematical correct version). A bitt more cleaner now. Throught I 'm not shure if angles and z value is correct. 2D slice on z=0 looks as large scale structures of universe (or pics on washing powders), but 3D is much behind, what sugest that it's the z. And 2D pic is without scale. Alsou with some spin values y=0 plane are discontinuate. Any suggestions on z or his brute force renderous majesty of fragmentarium?
http://en.wikipedia.org/wiki/Gravitational_wave (http://en.wikipedia.org/wiki/Gravitational_wave)
radius = sqrt( sqr(zx)+sqr(zy)+sqr(zz) );
angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(zx+1i*zz);
//(angz=(atan2(zx+1i*zz))^2; ? ? ?)
zx= cos(frequency*radius+ spin*angx) *radius*scale+Cx;
zy= sin(frequency*radius+ spin*angy) *radius*scale+Cy;
zz= cos(frequency*radius+ spin*angz) *radius*scale+Cz;
with frequency, spin and scale being changable parameters.
(http://s1.postimage.org/nyjvykxgf/EOG_sin_4.jpg)
(http://s9.postimage.org/6ycizofvz/EOG_sin_5.jpg)
(http://s9.postimage.org/6pey9r33j/EOG_sin_7.jpg)
In plane z=0 this should look somewhat so. As it do.
(http://upload.wikimedia.org/wikipedia/commons/b/b8/Wavy.gif)
? ? ?
(http://upload.wikimedia.org/math/3/d/9/3d9be0663baddbc7b8fd531d6b20be65.png)
(http://upload.wikimedia.org/math/4/2/a/42a7f53129a3cf3836564daa1b1160cd.png)
It seems that angz=(atan2(zx+1i*zz))^2; generates the cleanest picture.
(http://s2.postimage.org/qzshbnpvt/EOG_angzsquared_2.jpg)
(http://s1.postimage.org/yk8t7pa3j/EOG_angzsquared_1.jpg)
http://en.wikipedia.org/wiki/Gravitational_wave (http://en.wikipedia.org/wiki/Gravitational_wave)
radius = sqrt( sqr(zx)+sqr(zy)+sqr(zz) );
angx=atan2(radius+1i*zx);
angy=atan2(zx+1i*zy);
angz=atan2(zx+1i*zz);
//(angz=(atan2(zx+1i*zz))^2; ? ? ?)
zx= cos(frequency*radius+ spin*angx) *radius*scale+Cx;
zy= sin(frequency*radius+ spin*angy) *radius*scale+Cy;
zz= cos(frequency*radius+ spin*angz) *radius*scale+Cz;
with frequency, spin and scale being changable parameters.
(http://s1.postimage.org/nyjvykxgf/EOG_sin_4.jpg)
(http://s9.postimage.org/6ycizofvz/EOG_sin_5.jpg)
(http://s9.postimage.org/6pey9r33j/EOG_sin_7.jpg)
In plane z=0 this should look somewhat so. As it do.
(http://upload.wikimedia.org/wikipedia/commons/b/b8/Wavy.gif)
? ? ?
(http://upload.wikimedia.org/math/3/d/9/3d9be0663baddbc7b8fd531d6b20be65.png)
(http://upload.wikimedia.org/math/4/2/a/42a7f53129a3cf3836564daa1b1160cd.png)
It seems that angz=(atan2(zx+1i*zz))^2; generates the cleanest picture.
(http://s2.postimage.org/qzshbnpvt/EOG_angzsquared_2.jpg)
(http://s1.postimage.org/yk8t7pa3j/EOG_angzsquared_1.jpg)
Title: Re: End of Greatness
Post by: kram1032 on January 03, 2013, 04:14:02 AM
If you want to go after those amplitude formulae, I suggest you collect all constants into one big term.
For instance, G is the gravitational constant and c is the speed of light. You might as well set both to 1 or you can artistically modify them.
R, if you want to keep it correct, would be a parameter giving the distance of the camera from the center of mass of the two orbiting mass - points. For a steady camera, this is a constant.
r is the distance between the two objects in question. This is another constant
gives the angle between the observer and the plane of the rotation, starting from the center of mass.
You have to slightly transform that angle so it becomes what you'd usually think of as spherical coordinates.
is the angular velocity. It's how fast the two masses orbit each other.
\cos\left(2\omega\left(t-R\right)\right)<br />h_\times = -\frac{2a}{R} \left(\cos\theta\right)\sin\left(2\omega\left(t-R\right)\right))
contain all the constants, like the masses and the gravitational constant.
This is essentially, how easy those formulae can get. The dependency of the amplitude in the z-direction is governed by. In the x-y-plane, the wave's amplitude is radially symmetric and given by
\right)<br />h_\times = -\frac{2a}{R} \sin\left(2\omega\left(t-R\right)\right))
Note how this amplitude drops over distance and how the frequency changes both over distance and time, increasing with time, decreasing with distance.
On the point where the distance equals the time, the frequency becomes 0. This is simply where you travel away from the center of mass at the speed of light.
If you actually want to use this for producing 3D static fractals, you probably don't want a time dependency or an amplitude drop. So in that case, you should modify those formulae to:
\cos\left(2\omega R \right)<br />h_\times = \cos\left(\theta\right)\sin\left(2\omega R \right)<br />)
In this geometric setup,=sqrt{R^2-z^2})
, so =z)
and may only vary between 
.
Essentially you could just use (a 3D version of) my swirl transform and put the two formulae in front of it as an amplitude. For the most general state, I think you would just use a super-position of the two formulae:
\cos\left(2\omega R \right)\right)+c_2\left(\cos\left(\theta\right)\sin\left(2\omega R \right)\right))
where 
and 
should probably be normalized.
I'm not entirely sure if that's how you're supposed to use them though. You could always just try to use one of them. The difference apparently is how the gravitational waves would distort masses:
versus 
(http://upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_PlusPolarization.gif)(http://upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_CrossPolarization.gif)
Technically, the two could also be superposed out of phase (e.g. with different times, rather than both synced up), which would cause more complex patterns.
For instance, G is the gravitational constant and c is the speed of light. You might as well set both to 1 or you can artistically modify them.
R, if you want to keep it correct, would be a parameter giving the distance of the camera from the center of mass of the two orbiting mass - points. For a steady camera, this is a constant.
r is the distance between the two objects in question. This is another constant
You have to slightly transform that angle so it becomes what you'd usually think of as spherical coordinates.
This is essentially, how easy those formulae can get. The dependency of the amplitude in the z-direction is governed by
Note how this amplitude drops over distance and how the frequency changes both over distance and time, increasing with time, decreasing with distance.
On the point where the distance equals the time, the frequency becomes 0. This is simply where you travel away from the center of mass at the speed of light.
If you actually want to use this for producing 3D static fractals, you probably don't want a time dependency or an amplitude drop. So in that case, you should modify those formulae to:
In this geometric setup,
Essentially you could just use (a 3D version of) my swirl transform and put the two formulae in front of it as an amplitude. For the most general state, I think you would just use a super-position of the two formulae:
I'm not entirely sure if that's how you're supposed to use them though. You could always just try to use one of them. The difference apparently is how the gravitational waves would distort masses:
(http://upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_PlusPolarization.gif)(http://upload.wikimedia.org/wikipedia/commons/b/b8/GravitationalWave_CrossPolarization.gif)
Technically, the two could also be superposed out of phase (e.g. with different times, rather than both synced up), which would cause more complex patterns.