Mixing 3D and 4D transforms

[David Makin]

Hi all, am just re-writing my 3D formula for UF so that it allows the user to plug-in the transform/s to be used on each iteration. I was going to stick to either 3D mode or 4D mode but thought maybe I should allow mixed modes ?
To render 4D fractals in 3D projected mode I normally use 2 of the 4 4D axes converted directly to 2 spatial axes and take the 3rd spatial axis as any straight line through the origin in the plane of the 2 remaining 4D axes.
This means that there are two obvious possibilities if allowing mixed 3D and 4D transforms:

1. On a 3D transform simply leave the 4th dimension values (z and dz) alone and modify the other 3, on a 4D transform simply use all 4 dimensions.

2. If you have 4D values for z and dz and the next transform is 3D then convert the values back to 3D based on your combined 3rd spatial axis, then perform the 3D transform, if the next transform is 4D then convert your 3D values back from the 3rd spatial axis to the 2D values for the 4D transform.

Has anyone tried this ?
Anyone any thoughts on whether the distance estimation (particularly the analytical) would hold up using either of the above methods ?

[kram1032]

I might be misunderstanding something here, but should be any 3D-transfrom just a special case of all the possible 4D-transforms? (And likewise any 2D transform a special case of possible 3D transforms)

You could treat a 3D transform on a 3D object as a 4D transform that only affects three of four perpenticular axes... (Not necessarily the main axes but any rotated set of three perpenticular axes in a 4D hyperspace...)

Or can't you? I'm not actually sure^^

[reesej2]

That certainly sounds interesting. How do you intend on converting BACK from 3D to 4D? It seems like when you combine the axes the information is going to be lost.

[David Makin]

That certainly sounds interesting. How do you intend on converting BACK from 3D to 4D? It seems like when you combine the axes the information is going to be lost.

In case 2. 3D to 4D the same way as you convert the spatial coords to 4D, i.e. treating one of the 3D axes as a line in the plane of 2 of the 4D axes (and vice versa for 4D to 3D).
Or as I suggested in case 1. where the 3D transforms just ignore the 4th dimension

[David Makin]

I might be misunderstanding something here, but should be any 3D-transfrom just a special case of all the possible 4D-transforms? (And likewise any 2D transform a special case of possible 3D transforms)

You could treat a 3D transform on a 3D object as a 4D transform that only affects three of four perpenticular axes... (Not necessarily the main axes but any rotated set of three perpenticular axes in a 4D hyperspace...)

Or can't you? I'm not actually sure^^

You can, and that's what I suggested in case 1, but I also suggested the case where you convert from 3D to 4D (and back) using essentially the same method as is normally used converting 3D spatial coords to a 4D value (where one of the 3D axes is a line in the plane of 2 of the 4D axes).
In either case I'm not sure how it would affect the analytical DE, though I guess just ignoring the 4th dimension in 3D transforms would probably mean that analytical DE would just work normally

[kram1032]

Ah, I see.
Both ways sound fine to me...
"just" make it possible to choose three aribitary axes for 3D transforms in 4D.
Hmmm.... maybe even generalize it to axes which are not neccesarily normal on each other? That wouldn't be a usual 3D transform of course but it might yield interesting effects in some cases...

[M Benesi]

1. On a 3D transform simply leave the 4th dimension values (z and dz) alone and modify the other 3, on a 4D transform simply use all 4 dimensions.

2. If you have 4D values for z and dz and the next transform is 3D then convert the values back to 3D based on your combined 3rd spatial axis, then perform the 3D transform, if the next transform is 4D then convert your 3D values back from the 3rd spatial axis to the 2D values for the 4D transform.

Has anyone tried this ?

  I've done lots of higher dimensional (4+) renders and animations, no transforms.  I've experimented with various things (setting the higher dimension to either sqrt(x^2+y^2+z^2) or sqrt (...) * some value, other magnitude combinations) and found... that lots of different techniques give pleasing (to me) results.  <--  not too helpful...