Spherical Coords w/ Converging Latitudes

[BradC]

On some thread somewhere somebody mentioned something like this. This formula is similar to the normal Mandelbulb, except that the latitude coordinate lines converge toward the poles as fast as the longitude lines converge, making the longitude & latitude transformations together a conformal map. (Notice how the latitude & longitude lines continue to form squares even near the poles, rather than long skinny rectangles as with the normal latitude spacing.) If you include r however, the coordinate system is not conformal. In this system, the equator is at phi = 0 and the poles are at phi = +-infinity. To square a point I squared the radius and doubled theta and phi as usual.

Unfortunately, the result isn't fractal except in the xy-plane, but I'm posting it anyway because it looks interesting, like an inflated Mandelbrot set. Here's degree 7:

Here's degree 2:

[kram1032]

Did you use a higher iteration count on this one too?
Because the 2D-Version kinda looks like it would get a lot of new details later...

[BradC]

I used a max iteration count of 64, but I was also bailing out based on a distance estimation, which probably removed some of the details. Also notice on the degree 7 version how the "seahorse valleys" look like they protrude into the balloon so they get covered up and you can't see the full depth of the valleys. This might be happening on the degree 2 version too. Also I don't think the distance estimation formula I'm using works very well with these fractals.

[kram1032]

ah, wait...
the poles are at phi=+/-inf?
what happens if you just take a slice on where phi = +/- 1 or +/- pi or something?

Actually, this might not be too far from what we want...
the bulbs "just" need to be spherified and "copied" over the surface...

It looks like if you want to do a balloon right now

Try to not use a distance esitmation if that doesn't totally kill rendertimes and also try to zoom into the searhorse valley, now seemingly being a hole...

also try to do phi = 2/phi (so, inverting current phi and multiplying that by 2) as phi = infinity probably wouldn't be reached otherwise... just some ideas to try out