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That does not easily work, because the sphere and the flat plane are not really ... compatible (the buzzword here is "topology"). Riemann had to add "infinity" as one more point to the set, otherwise his sphere would have had a tiny tiny hole at its pole. The sphere needs that one extra point to be complete, but the flat plane does not really have room for that one point. That's why the added point does not have coordinates in the flat plane; it needs to be out of reach of the usual (x, y) number pairs. That's why it is called the "infinitely distant" point, literally beyond the flat coordinates.

Or in other words, when you try to lower the pole of Riemann's sphere to lower angles, the sphere will have a hole and be incomplete. There would be points for which no coordinates can be easily defined. So you would be unable to do any fractal iterations with those points.


My other point about morphing between the classical Mandelbulb and the monopolar fractal is a bit trickier than I thought. There is no obvious way to parameterize the circular grid lines consistently. I noticed this when I tried to do a sketch with a north pole and an east pole. It is possible to do that, but the sketch would be somewhat arbitrary with respect to the distances between adjacent grid lines.


About your renderer: this was not an attempt to bait you into giving your code away. I was just impressed how quickly you were able to render an arbitrary formula in such high quality. I am currently tinkering with my own pre-pre-alpha version of a low quality renderer. It can already render flat outlines of 3D fractal shapes. It will be terribly slow and imprecise, but it's fun to go my own way, and see the shapes slowly materializing out of the fog, so to speak ... :-)

I talked Jessy into letting me host his animation on my webspace. The video shows monopolar bulbs with a smoothly varying exponent from 2 to 10. This is an experiment to see what my humble virtual server can handle, so the video is linked from nowhere but here. Have a look at: http://www.vectorizer.org/guesthouse/RiePow1to10.AVI.

As far as I can tell, the displaced origin causes the grid cells of the "Riemandelbulb" to shift around a bit, without changing their underlying shape. It doesn't change the nature of the fractal, but breaks the perfect symmetry. I cannot render good pictures yet ... trying to shade a fractal surface is a thankless job. :-)

I guess I really need to find out more about this idea of moving the poles independently. Morphing between the original Mandelbulb and the Riemandelbulb should lead to more interesting shapes. But for now I am wrestling with rendering algorithms.

Oh and the "little anim" is quite striking. This thing really seems to differ from the Mandelbulb in substantial ways. Thanks for taking the time to render this!


Edit: another thing. I need a better name for the monopolar fractal. How does "Riemandelettuce" sound? :-)

Riemandelsche Ananas? Haha, nice

Wow, for some reason I don't get it... Could you post pseudocode for normal? Thanks.

A sidenote: I decided to cheat for my rendering experiments. I will separate the steps of fractal creation, distance estimation, and rendering/lighting from each other. I know this approach has downsides (for example, you cannot efficiently animate changing parameters of the formula), but overall it seems to be a good trade-off.

The plan is
1. to fill a 3D volume with just the inside/outside information of any arbitrary fractal formula (just one bit per voxel)

2. do a general distance transform on that (only needs to be very accurate near the fractal border, so one byte per voxel should do)

3. trace rays through the volume of distances to render isosurfaces

The hard part was number 2, but I made very good progress on that. One 1000x1000 slice takes less than 0.5 seconds to transform with Saito's algorithm, and extending the method to three dimensions is straight forward. So I should be able to obtain the exact distance transform of a 1000^3 volume in less than 1000 seconds on one processor core. (That's why my renderer will only be good for flying around static objects, but that's my first goal.)

I already have a basic raytracer that uses second order spline interpolation of the voxel data for smooth isosurfaces and consistent surface normals. I hope I can speed it up significantly using the exact distance information.


My renderings won't be so beautiful, but I should be able to produce fly around animations of any weird fractal formula, no matter if an analytical distance estimate even exists. And I would be able to render inside views. Main memory is the limiting factor of the basic method as outlined here - a 1024^3 volume needs at least a gigabyte of RAM during rendering - but if this works at all, it can always be extended with more efficient data structures like an octree.

Hobold, in my wishlist is also a flymode, or as i would implement it: a slow walking mode 
Doing some basic OpenGL that is not too difficult and calculating the surface in the near surrounding via CPU.

But i have to finish the current project before...

@Timeroot: sorry for hijacking your thread. I hope to return to the original journey, exploring more exotic coordinate systems. I try to keep my planned renderer generic and independent of specific formulas for that very reason.

Your sketch of "quadrupolar" coordinates ... I am trying to picture them. It seems to me that what you are doing is to drag out each pole into a finite line segment. The ending points of such a line segment would then be the focal points of "concentric" elliptical grid lines. All this projected onto a sphere, so the grid lines are really polynomial curves of order four or higher. This will be complicated. But by varying lengths and orientations of the northern and the southern line segment each, you would be able to break the symmetry of the standard spherical coordinates. This could well lead to more variety of shapes.