I've been reading this article:

http://spacesymmetrystructure.wordpress.com/rheotomic-surfaces/#comment-3666

I was wondering how hard it would be to reproduce his work in POVray.

I'm especially interested in reproducing the curves in this image:

http://spacesymmetrystructure.files.wordpress.com/2009/02/diaglines-converted.jpg?w=510&h=510

as well as plotting the circles in this image:

http://spacesymmetrystructure.files.wordpress.com/2009/02/circles-converted.jpg?w=510

What do you guys think?

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The lines in the grid look like gravitational distortions of space-time.
I'm wondering if the equations are similar to satellites and black holes.
especially those lines where they curve in, and then curve out again, near the
"empty" spaces.
Follow a line from the edge and picture a falling object getting pulled into
orbit around an initally closer gravity well, then getting grabbed by another
one and sucked into that...

You might want to hit up some people on the math forums - mathematica, etc.
Real whiz-kids on there...

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"Bald Eagle" <cre### [at] netscapenet> wrote:
> The lines in the grid look like gravitational distortions of space-time.
> I'm wondering if the equations are similar to satellites and black holes.
> especially those lines where they curve in, and then curve out again, near the
> "empty" spaces.
> Follow a line from the edge and picture a falling object getting pulled into
> orbit around an initally closer gravity well, then getting grabbed by another
> one and sucked into that...
>
> You might want to hit up some people on the math forums - mathematica, etc.
> Real whiz-kids on there...

Yes, it looks like it's based on the formula for gravitational fields (or maybe
electrical fields). I think the pictures are very pretty.


Mike

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> Yes, it looks like it's based on the formula for gravitational fields (or maybe
> electrical fields). I think the pictures are very pretty.

I was thinking it looked like magnetic lines of force - but it was the holes
that made me think using a black-hole warp or something might lead you
somewhere..
At least as a first approximation.

It looks deceptively simple on first glance...  and yes, wicked cool!

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http://www.grasshopper3d.com/profiles/blogs/rheotomic-surfaces-and-flowline-generation-tool

http://www.grasshopper3d.com/forum/topics/set-boundary-in-rheotomic-surface-definition

http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html
(!)

Try this:
http://api.ning.com/files/UA0agJ6Ue-7ZRzo7DWX8vVAY0014CD0J70kYxzRLC08kll8PoAP48Rk2EFak*tJOW9WEdwMli40BZPA-eU3v2g__/flow
line_surface.png

http://www.openprocessing.org/sketch/17344

http://www.openprocessing.org/sketch/34033

http://www.mai.liu.se/~halun/complex/domain_coloring-unicode.html

http://vimeo.com/3356808

http://vimeo.com/3029664

http://vimeo.com/37060860

"The surfaces are generated first as complex heightfields and the flowlines come
for free as the contours of those surfaces!"

http://www.grasshopper3d.com/photo/principal-stress-grid?commentId=2985220%3AComment%3A266221&xg_source=activity

http://geometrygym.blogspot.com/

http://en.wikipedia.org/wiki/Einstein_field_equations

http://www.cgal.org/Manual/latest/doc_html/cgal_manual/packages.html#part_XII

http://www.mai.liu.se/~halun/complex/

http://graphics.stanford.edu/~niloy/research/constrained_mesh/paper_docs/constrainedMesh_sigA_11.pdf

http://design.epfl.ch/organicites/2010b/tag/mid-term/feed


.... and plenty more just from Googling "equations rheotomic surfaces"

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Wow, thanks for all the links! Unfortunately, I can't make heads or tails of
anything. It's more math than I can handle. :(


Mike



"Bald Eagle" <cre### [at] netscapenet> wrote:
>
http://www.grasshopper3d.com/profiles/blogs/rheotomic-surfaces-and-flowline-generation-tool
>
>
http://www.grasshopper3d.com/forum/topics/set-boundary-in-rheotomic-surface-definition
>
> http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html
> (!)
>
> Try this:
>
http://api.ning.com/files/UA0agJ6Ue-7ZRzo7DWX8vVAY0014CD0J70kYxzRLC08kll8PoAP48Rk2EFak*tJOW9WEdwMli40BZPA-eU3v2g__/fl
ow
> line_surface.png
>
> http://www.openprocessing.org/sketch/17344
>
> http://www.openprocessing.org/sketch/34033
>
> http://www.mai.liu.se/~halun/complex/domain_coloring-unicode.html
>
> http://vimeo.com/3356808
>
> http://vimeo.com/3029664
>
> http://vimeo.com/37060860
>
> "The surfaces are generated first as complex heightfields and the flowlines come
> for free as the contours of those surfaces!"
>
>
http://www.grasshopper3d.com/photo/principal-stress-grid?commentId=2985220%3AComment%3A266221&xg_source=activity
>
> http://geometrygym.blogspot.com/
>
> http://en.wikipedia.org/wiki/Einstein_field_equations
>
> http://www.cgal.org/Manual/latest/doc_html/cgal_manual/packages.html#part_XII
>
> http://www.mai.liu.se/~halun/complex/
>
>
http://graphics.stanford.edu/~niloy/research/constrained_mesh/paper_docs/constrainedMesh_sigA_11.pdf
>
> http://design.epfl.ch/organicites/2010b/tag/mid-term/feed
>
>
> .... and plenty more just from Googling "equations rheotomic surfaces"

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Yeah, I know the feeling.
I've got like 7-8 pages of differential equations I took as notes while trying
to figure out the Finite Element Method.

Start with something simple.

Try some heighfields, some basic grids / meshes and try some deformations and
transformations.

http://www.f-lohmueller.de/pov_tut/pov__eng.htm
Try looking through this stuff and Googling "POV-Ray + ....."
Then take baby steps until whatever you're doing approaches your goal.

I find this interesting as well, but right now I'm in the middle of 2 or 3
sh*tstorms IRL, and it'll likely be about 6 months before I can resume a lot of
this.

Try some grids / checkers and apply some warps, and I think that will get you
far, fast.  Watch the videos - they're illuminating.

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One of the Java applets you linked to doesn't seem too complicated. I'll try and
reverse engineer it.

I want to use the equations themselves instead of relying on existing POV
primitives and patterns since I want to use them outside of POV too.


Mike


"Bald Eagle" <cre### [at] netscapenet> wrote:
> Yeah, I know the feeling.
> I've got like 7-8 pages of differential equations I took as notes while trying
> to figure out the Finite Element Method.
>
> Start with something simple.
>
> Try some heighfields, some basic grids / meshes and try some deformations and
> transformations.
>
> http://www.f-lohmueller.de/pov_tut/pov__eng.htm
> Try looking through this stuff and Googling "POV-Ray + ....."
> Then take baby steps until whatever you're doing approaches your goal.
>
> I find this interesting as well, but right now I'm in the middle of 2 or 3
> sh*tstorms IRL, and it'll likely be about 6 months before I can resume a lot of
> this.
>
> Try some grids / checkers and apply some warps, and I think that will get you
> far, fast.  Watch the videos - they're illuminating.

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Well then it seems to me that you'll need to get a firm grasp on the Laplacian
Operator "del" or "nabla".
http://en.wikipedia.org/wiki/Laplacian

Since it is used frequently in many fields of study, including "blob" and "edge
detection", you may wish to ask the folks who program the POV-Ray render engine,
look at the source code, talk to the brainiacs at http://math.stackexchange.com

http://math.stackexchange.com/questions/5076/what-does-upside-down-triangle-symbol-mean-in-this-problem

Check out Paul Nylander's page  www.bugman123.com

and basically play around with "systematizing" the operator.
What I mean by that is if you can find a correlation between where you start,
and where you want to go, you may be able to bypass 5 pages of math because you
can work out a little system to jump from
x^6 + 3x^5 + 9x^4 + 10x^3 + 32x^2 + 975.4x + 111.125 pi

to

6x^5 + 15x^4 + 36x^3 + 30x^2 + 64x + 975.4

in a heartbeat, without actually "doing" any "math".

Once you can systematize and parameterize the more complex equation into smaller
subsets of usuable and changeable fragments, you can then play around and
visualize what's going on with the remaining parts and further dissect the
equations until you have as close to a general solution as possible.

You should be able to bang out something workable in a spreadsheet, SDL,
Mathematica, Matlab, MathCad, etc.

In short, don't get yourself bogged down with understanding the fundamentals of
taking the above derivative if your real goal is to find a _practical_ solution
and USE the results of solving the Laplacian (or any other) complex operator.

If you use POV-Ray to solve / simulate what you want graphically, then you can
Trace() the elements of the grid, come up with some numerical solutions, and
sort of work backwards, or at least have numbers in had to guide you in
unraveling the equations since you can "see" if what you're doing with del this
and del that of x y and z is converging onto or diverging from what you already
know is purty darned close to the "real" mathematical solution.

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If you can reverse engineer Java applets, maybe this might help you some more:

http://virtualmathmuseum.org/ODE/index.html

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