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Author Topic: Polyhedrons, many many polyhedrons...  (Read 20939 times)
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kram1032
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« Reply #45 on: February 05, 2012, 05:21:07 PM »

Well, then it already uses stereographic projection, which was one of the two projections I suggested...
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Syntopia
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« Reply #46 on: February 05, 2012, 06:15:32 PM »

Knighty, as I understand your script, you take a point in 3D space and find out which point it corresponds to on a 3-sphere in 4D (using inverse stereographic mapping).

Then the vertices of the polychora are projected onto the 3-sphere (actually, I could imagine the vertices of a polychora are already located on a 3-sphere, so maybe just scaling?), and the DE distance is calculated (and the magical 'DD' function is applied to do some planar/spherical conversion - still need to figure out how that works).

I was thinking - is it possible to apply the same kind of projection to a 4D Quaternion Julia (or Mandelbrot) system? I tried using inverse stereographic projection, but as I understand it, this will only depict the intersection surface of a 3-sphere with the 4D Quaternion Julia system - I need to somehow project the points onto this 3-sphere, right? - but then the 3-sphere will get 'filled', I think...

I suspect that the boringness of these Quaternion systems (espically the Quaternion Mandelbrot), could by because of our simple slicing depiction. I don't know if anyone has tried other projections of these systems?

Finally, some of the Wikipedia pictures of polychora, are having straight lines, indicating a perspective(?) projection, rather than a stereographic projection. Anyone knows how this is achieved? Is it possible also to do using DE's?

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knighty
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« Reply #47 on: February 05, 2012, 08:57:16 PM »

Beside the projection to the 3-sphere and the "magic" formula the process is exacly the same as for 3D-polyhedron.
As you said, the 3d point is projected on the unit sphere using inverse stereographic projection.
The maths behind the "magic" formula is quite simple (see picture below). It's indeed used to convert distance on the sphere (3-sphere) into distance on the plane (3D-space). This conversion is essentially a 2D problem:
The given point Z is projected onto the unit sphere giving Z'. Say, "alpha" is the angle between Z' and the x-axis.
Say we know the spherical distance beween Z' and a point on the sphere ("delta" in the figure). The spherical distance is simply the angle beween Z' and that point.
Now we construct a point P' on the sphere which angle is "alpha"+"delta". Then we project P' back on the plane and get P.
The distance between P and Z is the distance estimate.

The cosine of the angle between Z' and a vertex V of the polychora is simply the dot product of Z' and V. For the segments we have to find the closesest point to Z', project it onto the sphere (by normalizing it) then compute the dot product. the sine is a little more involved because we don't have vector product in 4D. Using sqrt(1-cos▓(delta)) is not accurate enought.

The renderings in wikipedia of polychora with  straight lines is maybe done with the classic  cheesy method. having the set of vertices and segments. Project the vertices onto 3D space using some appropriate projection (ideally one that shows most of the symmetries) Then render it as usuall. As I said in a previous post it is possible to use DE but it would be very slow (that also may answer kram's question). It's applicable to any well behaving 4D distance field and is just like raymarching... but it's a double raymarching:
From the point of 3D space shoot a ray into 4D (much exactly what we do in 3D but in 4D: the screen space becomes a box wink). Find the minimal distance along that ray. This is the distance we will use to marche in 3D.

AFAIK, the boringness grin of quaternionic fractals comes from the so called j,k equivalence, not from the projection. In particular quaternionic madelbrot set is simply a rotated 2D madelbrot set around the real axis in j and k directions.


* magicf.JPG (62.76 KB, 693x555 - viewed 167 times.)
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Syntopia
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« Reply #48 on: February 05, 2012, 11:51:13 PM »

Thanks, Knighty - I think I got the idea of the 4D polychora DE now. Well... there is still the hyperboloid / special relativity stuff, which sounds really interesting, but I have to post-pone it till I get some more time to pursue it.

Btw, I've posted some images of the regular convex polychora here: http://blog.hvidtfeldts.net/index.php/2012/02/distance-estimated-polychora/
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knighty
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« Reply #49 on: February 06, 2012, 09:36:28 AM »

Me too, I'll have to find some time to implement hyperbolic space tilings.

For the segments we have to find the closesest point to Z', project it onto the sphere (by normalizing it) then compute the dot product.
Keep in mind that that the closest point to Z' of the projection of the segment on the sphere is not the projection on the sphere of the nearest point on the segment to Z'. Instead, it is the projection on the sphere of the orthogonal projection of Z' on the plane formed by the polything vertex and the (mirror) plane which is perpendicular to the segment.

Also, this method is unfortunately not applicable IMHO to the snub polyhedra/chora.

BTW, here is a great article that explains much better how to render stuff in non euclidean spaces and their geometry :
Jeff Weeks,"real-time rendering in curved spaces",Computer Graphics and Applications, IEEE, Nov/Dec 2002.
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KRAFTWERK
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« Reply #50 on: February 06, 2012, 10:18:59 AM »

I just love this! Mind blowing animations!
Keep up the good work knighty & subblue & the rest of you!
I must try out fragmentarium. I feel at home from the pixelbender days.. smiley
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #51 on: February 06, 2012, 07:04:50 PM »

I have implemented the most simple Knighty f&c scripts and WOW how wonderful!

Added too a sphere ... The stars remembered me some videogame bombs grin so much fun!


* tyytr.JPG (95.51 KB, 1042x979 - viewed 178 times.)
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No sweat, guardian of wisdom!
knighty
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« Reply #52 on: February 10, 2012, 08:58:53 PM »

Hi,
Thank you Kraftwerk. Fragmentarium can also generate (very) big pictures by dividing in tiles and assembling them for you. Very fast and comfortable ;o). It would be nice to have this feature in MB3D and mandelbulber.  evil

I have implemented the most simple Knighty f&c scripts and WOW how wonderful!

Added too a sphere ... The stars remembered me some videogame bombs grin so much fun!

Well done!  smiley

I didn't like the parametrisation in that script so I've rewritten it in order to have more predictable results + some enhancements.

I've also fixed some little bugs in the previous scripts.

* Poly-frags.zip (7.48 KB - downloaded 122 times.)

* starPoly.jpg (75.4 KB, 720x605 - viewed 193 times.)

* out-2.jpg (87.46 KB, 957x712 - viewed 165 times.)
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marius
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« Reply #53 on: February 10, 2012, 09:26:24 PM »

I didn't like the parametrisation in that script so I've rewritten it in order to have more predictable results + some enhancements.

I've also fixed some little bugs in the previous scripts.

Fantastic! These make great shapes to have tumble around in a larger fractal, eiffie style ;-)
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knighty
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« Reply #54 on: February 12, 2012, 08:53:12 PM »

  wink

Someone have already done a briliant hyperbolic space navi on GPU (there are many other variation on the site)  hurt. I'll have to do something better  tease.
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #55 on: February 12, 2012, 09:44:27 PM »

Go hero gooo cheesy
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No sweat, guardian of wisdom!
marius
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« Reply #56 on: February 12, 2012, 10:30:49 PM »

  wink

Someone have already done a briliant hyperbolic space navi on GPU (there are many other variation on the site)  hurt. I'll have to do something better  tease.

Indeed. Thanks for reminding me of glsl sandbox. Very cool and instructive stuff there.
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #57 on: February 14, 2012, 04:21:00 PM »

Found an interesting image gallery

http://www.flickr.com/photos/fdecomite/sets/72157613498998540/

 Azn
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subblue
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« Reply #58 on: February 14, 2012, 04:31:58 PM »

Ahh yes, fdecomite, has done some very interesting work. I've been following his stuff for some time smiley
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www.subblue.com - a blog exploring mathematical and generative graphics
knighty
Fractal Iambus
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« Reply #59 on: March 10, 2012, 08:48:51 PM »

Better late than never. Here is a 3D hyperbolic tesselation fragmentarium script. Just *-3-*-5-*-3-* Coxeter symmetry group for now. Others will come later.


* out.jpg (253.74 KB, 736x668 - viewed 150 times.)
* hyperbolic-tesselation-3-5-3-colored-02.frag (7.05 KB - downloaded 103 times.)
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