Supermodulus radius generating superelipse shape

[Alef]

I had it forwarded to Luca and but I think this should be shared to public maybe it could be usefull. Calculation of radius is key element in Mandelbulb, Mandelbox and in lot of other formulas and the radius could be calculated like this:

radius := abs(x) + abs(y) + abs (z); - diamond aka cube turned by 45.
radius := sqrt (x^2 + y^2 + z^2); - sphere.
radius := ( x^4 + y^4 +z^4 ) ^ 1/4 - pillow.

All of these versions of radiuses must work in spherefold. I put the power 4 radius in IQ bulb and got "Boxbulb". Jet the abs based function generated mandelbulb was strange.

Both diamond and pillow is in the catogory of superelipse - the shape of iPhones. If it creates superelipse shape then maybe it is supermodulus.
https://en.wikipedia.org/wiki/Superellipse

This function can be parametrised. But I would not advise to do it in 3D fractals becouse that will bbeeee ttttooooo sssllllooowwwww.

radius := ( abs(x)^power + abs(y)^power + abs(z)^power )^(1/power)

Arbitrary power is slow function.






https://en.wikipedia.org/wiki/Superquadrics



I found this thing long time ago in 2D with mandelbrot + unit vector addition.
z=z^2+c
z=z + factor* z/|z|

Preferable factor = -0.5 when it generates hollow circles. With superelipse it generates hollows like these shapes.

[Alef]

It is:
https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions

I have no idea what that article talks about   But it seems that I have rediscovered it

For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by
= ( abs(x)^power + abs(y)^power + abs(z)^power )^(1/power)

Rounded box DIFS could be something related to it.
https://en.wikipedia.org/wiki/Superegg

[mclarekin]

Thanks Alef,
I will try this out.


the pillow formula is exactly what I had just been looking for today

[CozyG]

...
https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions
I have no idea what that article talks about   But it seems that I have rediscovered it
For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by
= ( abs(x)^power + abs(y)^power + abs(z)^power )^(1/power)

The easiest way to understand p-norms (at least for me) is to consider them as different ways to measure distance. For p=2 (the 2-norm), in 2D you get distance = sqrt(x^2 + y^2), which is the Euclidean distance we consider the normal way to measure distance in 2D space. But consider the 1-norm (when p=1): distance = abs(x) + abs(y). The 1-norm is also called the taxicab distance, in reference to a taxicab travelling along an idealized grid of streets. In a taxicab, the important measure (for both driver and passenger) is not the Euclidean distance from point A to point B "as the crow flies", but the taxicab distance which will be the sum of the distances along each street. So p-norms are different but potentially valid ways of measuring distance.

There's been some interesting work recently on applying p-norms to circle and sphere inversions. In a paper by Ramirez et al this is referred to as p-circle inversion, and in a paper by Gdawiec it is referred to as pseudo-inversion. Based on those papers I've done a little bit of work recently with p-norms while developing enhanced circle inversion variations for JWildfire.

In Mandelbulb3D I think ABoxVaryScale2, with the right settings for RPow and XYZPow, essentially uses a p-norm measure for the distance between the iterated point and the origin of the sphere of inversion.

I do like the idea of applying p-norm measurements as options to other formulas. I'll add it in soon as an option for the 2D Mandelbox code I'm developing for Fragmentarium over in the thread "Generalized shape inversion applied to the Mandelbox"

PS Your initial post in this thread motivated me to add a superellipse as one of the shape options for that code

[hobold]

The easiest way to understand p-norms (at least for me) is to consider them as different ways to measure distance.

You forgot to mention the punch line :

"A circle is the set of all points that are at a distance of [radius] from [circle center]."

So a specific way of measuring a distance between two places directly implies a corresponding kind of circle or sphere. (There are a few rules to be followed to ensure everything is logically consistent, but those rules are simple.)

[Alef]

One interesting feature of this is that superfomula is based on this. Superformula is patented.

I just did some thing where each axis had different powers. The only problam I don't know, what to use as mean value for inverse power - root. Geometric mean seems to work best but I 'm not shure is it anywhere correct. Alsou harmonic mean worked nicely but sometimes destroyed far away DE.

its like:
temp = abs(x)^k+abs(y)^l+abs(z)^m
rootpower= 1/ cubic root of ( k*l*m) // geometric mean of power values
radius = temp^rootpower

Of corse these functions are slow.
Not shure how to call this. I called it superseed becouse it looks like seed.


I was thinking that rounded box is something like this. It looks similar but wikipedia sayes it is a different shape. Have no idea about formula. A tought of rounded Abox (forgot the formula). Maybe it was one you mentioned.

[Alef]

CozyG:
You explained the only paragraph I understanded;)
At first I thought it was something like curved spaces hyperbolic space or spherical one. But it seems that this is different.

As I understand other transformations in Lp spaces must be different, too. Say translation if not simple addition or multiplication. Ideal would be a mandelbulb like city plan with angular trigonometry..

 ABoxVaryScale2 as described must be Lp - norm. It sayes  

More elaborate power expression for the radius of sphere inversion:
R = (x^P2+y^P2+z^P2)^P1

Throught implementation must be something different. At least it must be R = (|x|^P2+|y|^P2+|z|^P2)^P1  or else odd powers would not work.

RoundedBox difs looks simmilar of shape but is of different origin. But parameter "Border" is inverse of "p". In wikipedia was mentioned rounded square what is more easy to construate but have less easy formula.

So nice that there is Sci Hub site. So I can see what they are researching for free;)

I did power 30 mandelbulb in p=1. It is like aztec pyramids. Just that lower powers are ugly.

[Alef]

Inspired by superformula I made different curvatures of p in each axis. Maybe analogy could be Flat Earth with direction "up" being very different from directions "north" or "east". But realy it was meant to become a core for mandelbulb.box

Mean power of 3 p-s (sCurvatures) is calculated by geometric mean, and then inverted to root:
invpower= 1/ cubic root (sCurvatureX * sCurvatureY * sCurvatureZ)

Optimised:
invpower:=power(PowerX *PowerY *PowerZ*PowerW,-0.33333333333333333333);
(It needs a lot of 3s, this is visible. Numbers of 3s depends on Double or Float)

Then radius is calculated by
R= (abs(x)^sCurvatureX + abs(y)^sCurvatureY + abs(z)^sCurvatureZ)^invpower;

What I got I called a superseed becouse some combinations looks like seeds.
Then it is visialised by iterating radius in modified sigma function:

divisor:= 1/(Sigma+1);
temp:= (Sigma + R )*divisor;
(x,y,z) := temp*(x,y,z)

[Alef]

There is one important about these shape. p=3 or p=4 looks very advanced and streamlined. So they are favored by designers of cell phones. All cell phones are superelipses or
https://en.wikipedia.org/wiki/squircle



Alsou inter-city tourist buses are p=4 or p=3 shapes. Not shure does this add to their aerodinamics, as all tracks are cubes. But it certainly adds to their looks. Some of then looks like big ferrari.

[Alef]

In Apophysis and in flame fractals there are Fisheye transformation:

radius = square root of( x^2 + y^2)
temp = 2/(r+1)
(x,y) = temp*(x,y)
It generates scenery as fish eye lens but mathematicaly it is related to sigmoid function:
https://en.wikipedia.org/wiki/Sigmoid_function
(Not sigma, may mistake)
I think this is the obvious place were radius can be changed to p-norm.