angular singularity -> whipped cream?

[cKleinhuis]

hi all,


dunno if it is discussed, but when waiting for the bus this day, i was thinking about the singularity problem, when working with euler
angles, we use just 2 angles, to describe a position on a sphere, but, is it sufficent ?!?!? dont we miss something ?
shouldnt be the second axe always perpendicular to the other, to maintain exact positioning on the sphere ?

so, i think the whipped cream problem occurs due to singularity problems.... ( hope you know what i mean, i mean the fact that
when rotating an object around an axis, and then rotating it around another axis, it can happen that the resulting rotation is the same as before on the untransformed object with the other axe ... )

another idea i had, but is quite unevolved is the fact to use quaternions as description of the spherical coordinates

just some thoughts for the old year!

[twinbee]

Yeah, that's why I think the Mandelbulb has whipped cream too. Unfortunately it seems tricky if not impossible to remove the singularities because of the way spheres work in general. A while back, I tried looking at different spherical coord systems, but they each have their own issues (warping of coord space along certain parts of the sphere etc.).

[fractalrebel]

hi all,


dunno if it is discussed, but when waiting for the bus this day, i was thinking about the singularity problem, when working with euler
angles, we use just 2 angles, to describe a position on a sphere, but, is it sufficent ?!?!? dont we miss something ?
shouldnt be the second axe always perpendicular to the other, to maintain exact positioning on the sphere ?

so, i think the whipped cream problem occurs due to singularity problems.... ( hope you know what i mean, i mean the fact that
when rotating an object around an axis, and then rotating it around another axis, it can happen that the resulting rotation is the same as before on the untransformed object with the other axe ... )

another idea i had, but is quite unevolved is the fact to use quaternions as description of the spherical coordinates

just some thoughts for the old year!

This looks like and interesting area to pursue. 

[Paolo Bonzini]

another idea i had, but is quite unevolved is the fact to use quaternions as description of the spherical coordinates

I had written down how to do that but I have to fix some problems related to that (mostly notation, the basic idea is sound):

http://github.com/bonzini/mbulb/raw/master/mbulb.pdf

edit: done

[cKleinhuis]

thank you very much, right now i am starting a book projekt about fractal rendering in general, and
the mandelbulb in special ... certainly i will come back to you for certain informations, thank you

cheers

another idea i had, but is quite unevolved is the fact to use quaternions as description of the spherical coordinates

I had written down how to do that but I have to fix some problems related to that (mostly notation, the basic idea is sound):

http://github.com/bonzini/mbulb/raw/master/mbulb.pdf

edit: done

[cKleinhuis]

that is a fantastic paper, i like the definition of the familly of formulas that 'own' the mandelbrot

i am right before the "heavy" stuff part, i am enjoying reading it, i have to set up my development environment

but as i have understood correct, you have never used this formula for rendering the mandelbulb ( would be interested in if it
changes the look of the standard triplex algebra things ... )

and i see it that the family of formulas you describe is based on the simple fact that polar coordinates can be used for
exponentiation of complex numbers ? and this is why you developed the rotational axis parameter further and generalized
it, and you are thinking or hoping that any rotation axis would be better than another what is for sure is, that we now
have another nice set of parameters, which easily change the complete outlook of the fractal, and ALL of the share the
classical mandelbort set in their xy plane ... beautiful to imagine ....

have to implement that method, but first i have to "understand" more why you are doing it

[Paolo Bonzini]

Thanks. :-)

and you are thinking or hoping that any rotation axis would be better than another

It's certainly possible to develop math that would "favor" one definition over another, but I don't know enough to do that. :-)

but as i have understood correct, you have never used this formula for rendering the mandelbulb ( would be interested in if it
changes the look of the standard triplex algebra things) [...] have to implement that method, but first i have to "understand" more why you are doing it

It should not change anything.  As long as you do z=z^n+c, you are using triplex numbers in a way that is correct, it is just a shortcut through the "mathematically correct" description in the paper.  I'm not sure whether it's possible to apply the Mandelbulb idea to---for example---Newton's fractal, but that one would certainly have to be treated with care, and the concepts in the paper help being careful.

Just to quantify whether the triplex numbers had more magic or convenience in it.  As a programmer, I certainly saw the convenience, but at the same time inventing new mathematical objects didn't please me as much... I wanted to see if its properties (as well as its missing properties) could be justified using well-known things.

[cKleinhuis]

yes, a very good approach, i also have not enough mathematical understanding, what
i like about triplex algebra is that more and more operators seem to pop up, we have
division as far as i know ....

when i start implementing it, i try to "invent" new operators derived from the flame fractals,
so to say, i want some contracting function wich create interesting results ( flower, wave, swirl ... )
most of the flame transform functions easily expand to 3d and, should serve as new operators,
or as fn() replacement for the algebra ... just to create more interesting functions ....


but you are saying that the "whipped" cream effect is not solved with this method, but you
have generalized the rotation, and you have shortcutted or proved that it is commutative and
associative ( or did i miss something ) which opens great possibilities for new formulas ( lambda )

newtonian fractals are pain in the as, even in 2d, i believe it should be possible ....

 



 

[Paolo Bonzini]

yes, a very good approach, i also have not enough mathematical understanding, what
i like about triplex algebra is that more and more operators seem to pop up, we have
division as far as i know ....

You can define division on the quaternions (I think, I haven't worked out the details).

but you are saying that the "whipped" cream effect is not solved with this method, but you
have generalized the rotation, and you have shortcutted or proved that it is commutative and
associative ( or did i miss something ) which opens great possibilities for new formulas ( lambda )

It's commutative and associative, but still it does not mix well with addition.  If you want to add _and_ multiply, you have to sacrifice associativity.

[cKleinhuis]

what is the problem with addition ?!?!
is it because you need to convert it back and forth to vector/spherical coordinates?

[kram1032]

if you stick with the geometric definition, you could use dual quaternions for add and multiply.

Add basically shifts a 3vector which is possible by the dual quaternion while the rotations from multiplication are simply possible for the real quaternion as usual.

I tried to find a formula but I'm stuck at some part as the terms get quite complex.
In theory they should kind of simplify themselves nicely but the possibilities for simplifying I have are limited to wolfram alpha and a memory-limited TI-92 together with doing it manually...

And doing it manually easily introduces errors which are hard to correct afterwards....

Here is what I have for now with the work half done but manually and thus under circumstances incorrect...

The needed maths I got form there:
http://www.euclideanspace.com/maths/geometry/rotations/conversions/eulerToQuaternion/index.htm to convert Euler angles to Quaternions (I used heading and attitude as spherical coordinate angle equivalent, based on other content from over there and in the formulations simply assumed to use a general angle, rather than a fixed one....)

and

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/other/dualQuaternion/index.htm - where you can find the multiplication tables (I used the first of the two which seemed to be the correct one, if I understood the following descriptions) as well as the general transfom formula and the special case for a rotation followed by a translation.

As said all done manually so I can only hope that I did everything right....

There are two steps remaining: Simplifying and putting in the corresponding arguments. I hope this is clear... Wasn't easy to write it in a way that is overlookable and I guess I actually failed on that...

(rotationQuaternion)

      
      
      
      

(translation-Quaternion)

   
      
      
      


(location-quaternion)      

         
         
      
         



dual quaterion conjugate




general transform





rotation then translation






       
          
          
          









                   
                
                
                









   
   
   
   

[cKleinhuis]

oerx, awesome teXing

ehm, havent read it, but isnt that the last one would need for a complete algebra ( body, 2 operations, associative, commutative , each operation inversable) ?

[mrrgu]

Hi

A really good paper!!

I understand evrything up to the point there you set q = -vi ??
Can you explain that step please.

another idea i had, but is quite unevolved is the fact to use quaternions as description of the spherical coordinates

I had written down how to do that but I have to fix some problems related to that (mostly notation, the basic idea is sound):

http://github.com/bonzini/mbulb/raw/master/mbulb.pdf

edit: done

[kram1032]

Trifox: no idea^^
But if I figured it out correctly it should behave just like the Mbulb.... (However, as said, it's quite likely that I did some kinds of errors^^)

And about the texing: I'm not really familiar with LaTeX
I did all that in a .doc btw lol
Just copy, paste, throw tex tags around it and tweak until it works xD

[Paolo Bonzini]

I understand evrything up to the point there you set q = -vi ??

A 2D rotation is a rotation around the z-axis (i.e. "k").  Since I have to rotate the point 1i+0j+0k by 2theta and scale it by rho^2, i have to set q=rho * e^(k*theta)=x+ky.  If you expand -vi, it turns out that it is exactly x+ky.