Another take on Conformal Transforms

[hobold]

(Warning, abstract math ahead! This topic might not be for everyone.)

I have stumbled on a SIGGRAPH presentation about transformations that preserve 2D surface angle while deforming the surface within the surrounding 3-space. The paper's home page is http://users.cms.caltech.edu/~keenan/project_spinxform.html. The presentation video on youtube is a quick way to get a good impression on what the authors are doing.

I don't see any immediate use for us fractalists. I only have a nagging suspicion that maybe one could combat the whipped cream effect by embedding 3-space as a manifold in some higher dimension. Up there one could deform it conformally (to get more tunable fractal parameters, or a more interesting dynamic behaviour from iteration to iteration), then do some fractal iterations. Afterwards, conformally transform back to flat 3-space and finally render it.

I have not worked with this kind math before, so this could all be a false alarm. But maybe one of the other mathematicians here can make more sense of it than me.

[msltoe]

Interesting pictures. I don't understand the math.
I've often imagined the possibility that any 3-D deformation (Jacobian of a functional) could be a rotation in 5 dimensions, whereby stretching appears because we can't see the higher dimensions compensating. Then as the fractal iterations progress, the higher dimensions "appear" as more rotations are made.
I even tried coding this up:

 1) Compute the 3-D Jacobian, j of some arbitrary transformation, e.g. 3-D order 2 Mandelbulb.

 2) Find the closest unitary matrix fit in 5-dimensions: Place the 3x3 Jacobian in the top-left corner of a 5x5 matrix, J, and put two 1's along the 4th and 5th diagonal.
     Then compute R = (J' * J)^(-1/2) * J
     
 3) Apply this rotation, R, to a five dimensional vector (x y z w v).

 I don't remember anything "pretty" emerging, but maybe I was missing something.  The "fit" in 2) may not be optimal, for example.

[kram1032]

Hmmm... this seems like fairly simple math. When stuff like this emerges, you really wonder why it hasn't been found earlier already. But I'm not entirely sure if it simply expands to higher dimensions like that.... It might though. Often, once you find a way to get something that works in 2D to work in 3D, it's fairly easy to go up to arbitrary dimensionality. However, in this cases, it seems bound to quaternions. Since those are a 3D/4D special case (like the 2D/2D special case complex numbers), the first "easy" way to get there seems to be from octonions for 7D/8D...
Generalizing to any dimension might be a bit more involved by means of not having those special constructs. (Also, octonions might introduce problems in this line of reasoning...)

However, often in math, it turns out that you try to prove something and the way to that proof is extremely involved but then you end up with an extremely simple, beautiful formalism, highly obviously equivalent to what you already had for other cases so maybe...

[hobold]

However, often in math, it turns out that you try to prove something and the way to that proof is extremely involved but then you end up with an extremely simple, beautiful formalism, highly obviously equivalent to what you already had for other cases so maybe...

I call this the "Damn, why didn't I see this YEARS ago already!" effect ...

However, my own understanding of the methods is no where near that level. I just have this nagging suspicion that there is something in there which could be useful. Call it intuition. And don't rely on it!

[LhoghoNurbs]

Sometimes going to higher dimensions is not that straightforward. Especially when angles are involved.

Consider these two examples:

Example 1: Distance between two points in Euclidean space
in 2D it is sqrt(dx^2+dy^2)
in 3D it is sqrt(dx^2+dy^2+dz^2)
so it is pretty much easy to go from 2D to 3D to 4D to nD

Example 2: Conversion from "spherical" coordinates to Cartesian.
in 2D it is:  x=r*sin(a), y=r*cos(a)
in 3D it is:  x=r*sin(a)cos(b), y=r*cos(a)cos(b), z=r*sin(b)
so the shift from 2D to 3D is not that trivial.

[weshoke]

2) Find the closest unitary matrix fit in 5-dimensions: Place the 3x3 Jacobian in the top-left corner of a 5x5 matrix, J, and put two 1's along the 4th and 5th diagonal.
     Then compute R = (J' * J)^(-1/2) * J

Your idea is pretty close to what you'd need.  Instead of making the 5th diagonal 1, make it -1 and you have the 5D conformal model of 3D geometry.  There are some really interesting things you can do with it such as rotate translation operators and vice-versa.  There are also simple expressions of scaling, transversion, hyperbolic and spherical operations.

An easy to digest rundown of it can be found here:  see page 16 for the conformal GA matrix.
http://www.wolftype.com/versor/colapinto_masters_final_02.pdf

[Tglad]

In any number of dimensions more than 2, the only conformal transforms in Euclidian space are translation, rotation, scale and invert-reflect.
I don't see how converting up to more dimensions and converting down again can get around that fact.
I assume the quoted paper works because, whatever they are doing behind the scenes, they are only trying to keep a surface (2d) conformal.

Another idea is to work in Minkowski space, which gives you one extra transform to work with, a boost, which is the equivalent of a rotation between one space axis and the time axis: http://www.fractalforums.com/new-theories-and-research/relativistic-fractals/
You get what appears to be stretch... but it is stretch that actually does happen in real life, when you get close to the speed of light!

I wonder if we can go further into general relativity, and get some more transforms to work with.

[weshoke]

In any number of dimensions more than 2, the only conformal transforms in Euclidian space are translation, rotation, scale and invert-reflect.
I don't see how converting up to more dimensions and converting down again can get around that fact.
I assume the quoted paper works because, whatever they are doing behind the scenes, they are only trying to keep a surface (2d) conformal.

I don't think your assertion about dimensions > 2 is correct.  What about circular or spherical inversion?  The quoted paper is working in 3D space but applies equally to 2D.  For 3D Euclidean spaces, there is an "equivalent" space that is 5D called the 5D conformal geometric algebra.  It is Minkowski like you also mentioned since it's signature is (+, +, +, +, -).  It's not modeling space-time however, but provides a much simpler method for combining Euclidean conformal transformations.

There are a ton of resources describing this.  Just follow the links:
http://staff.science.uva.nl/~fontijne/phd.html
http://www.science.uva.nl/ga/tutorials/CGA/

[Syntopia]

I don't think your assertion about dimensions > 2 is correct.  What about circular or spherical inversion? 

I guess Inversions stricly speaking are anti-conformal (which is still good for fractals).
Liouville's theorem puts some strict limits for n>2:
http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29

[kram1032]

You know, the guy behind that paper has a YouTube channel...
http://www.youtube.com/user/keenancrane

Reading up on all of this a while ago; basically what he's doing is using Quaternions. So I'm not really sure what's exactly new about it.
As said, I'm surprised nobody else did so before. But either way, go through his videos. He has done some impressive work

[weshoke]

I guess Inversions stricly speaking are anti-conformal (which is still good for fractals).
Liouville's theorem puts some strict limits for n>2:
http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29

  The math on that page is way over my head.  I did find a reference in GA for computer science about the conformality (if that's a word) of spherical inversion. 

From the book:

The most elementary conformal transformation is the reflection in a unit sphere, called (spherical) inversion. As a versor, the spherical inversion in the unit sphere Σ around the origin involves the vector a = o– ∞/2, representing this sphere dually. The spherical reflection is performed by the versor product: <snip>

It is clear that this is not a Euclidean transformation, for ∞ is not preserved but interchanged with o (and weighted). Geometrically, this is understandable: the point at infinity reflects to the center of the sphere, and vice versa. The total result on a point x = Tx[ o ] is.

So it looks like the transformation is conformal but not Euclidean.

Here's a link, sorry for the "preview" nature of it.
http://my.safaribooksonline.com/9780123694652/B9780123694652500190_sec16-1

[Syntopia]

So it looks like the transformation is conformal but not Euclidean.

Inversions preserve angles and reverses orientation, making them anti-conformal (this is not the opposite of conformal, and some authors seem to include the anti-conformal transformation as conformal transformations, only requiring that angles are preserved):
http://en.wikipedia.org/wiki/Inversive_geometry#Anticonformal_mapping_property

[weshoke]

Ah!  thanks for clarifying.  I was not thinking about the term "anticonformal" in a mathematical/logical context.