Bour's Surface

From Mathematica - Code:

x(r, ϕ) = r cos(ϕ) - 1/2 r^2 cos(2 ϕ)
y(r, ϕ) = -r sin(ϕ) (r cos(ϕ) + 1)
z(r, ϕ) = 4/3 r^(3/2) cos((3 ϕ)/2)

Looks like it could lend itself to Math Functions XYZ or perhaps Math Nodes.

The closest I’ve seen to something comparable (code wise) is this from Instructables (using Blender):
(It’s regarding Mandelbulb, so, not sure how closely or if it correlates):

More on Bour’s Surface here:

image

(Maybe Procedural?)

what is your problem?

could be done with python script too !

mind you a bit more difficult
needs to make faces for the surfaces too!

happy bl

My apologies.

Forgot.

15 lashes with a python whip.

what do you mean with
15 lashes with a python whip ?

for the XYZ add on you need to use parametric equations
do you have these ?

thanks
happy bl

what would be the range for Radius and Theta ?

this would require double loop one over Radius and Theta
to make the 3D surfaces !

i could make a quick script using verts only
but faces requires more work

do you know how to write up python script ?

happy bl

Also I’m not up on my Greek.

Maybe this helps:

https://web.archive.org/web/20180618004315/http://paulbourke.net/geometry/bour/

I’m lost on r, t and v.

Also looks like the edges can either touch tangentially or overlap Enneper style.

Also this which might help:

https://demonstrations.wolfram.com/BoursMinimalSurfaceAndVariations/

that looks like some surface in the complex field
which makes it even more complicated to calculate and draw surfaces!

have not done a lot of surfaces in complex field
and in 3D complex don’t know how this work in blender LOL

not certain i got a lot of time to try something
may be later on

still possible using Python but could be easier using numpy

for a more elaborate script may be try in the python forum here or blend stack site

let us now if you get some answer for a script

note : for Bourn site it is probably not V but T
and this is in complex 3D field

thanks
happy bl

Hi unkerjay,

Not sure this helps but this is the best I could get with the XYZ math surface addon (from Blender 2.93.7).

These are the parametric equations:
X: a * cos(u) - 0.5 * a2 * cos(2*u)
Y: -a * sin(u) - 0.5 * a
2 * sin(2u)
Y: -a * sin(u) * (a * cos(u) + 1) # both Y equations yield similar results
Z: 4/3 * a**1.5 * cos(1.5
u)

U: -2pi to 2pi
V: -pi to pi

U wrap = on
V wrap = off
Close V = on

a = 1 : (radius)

Cheers,
Paul

Blender 3.0

If I plug in the equations as stated, I get this:

image

If I skip on X, Y, and Z everything after 0.5 or 1.5 respectively, I get no error, but, I don’t get your result either.

That tells me that there’s something in the way it’s constructed and interpreted that’s causing the problem.

I’ll try it on 2.93.7 and see what happens.

Blender 2.93.7 - same thing:

These equations don’t spit any errors, but, the result isn’t the same:

The text below isn’t the equations I entered:

a * cos(u) - 0.5 * a2 * cos(2u)
a * sin(u) - 0.5 * a2 * sin(2u)
4/3 * a**1.5 * cos(1.5*u)

Below is an image of the equations I entered:

image

I’m noticing that the equation I type in isn’t the same as what shows up when I hit the reply button.
I’m guessing the same is happening with the equations you entered, which would explain the problems.

Got the formulas right finally.

If you look at the image of the Bour’s surface though, compared to yours, yours has the Center vertex sitting low.

The example at the top has the Center vertex centered on the circle.

Yours is off center. Other than that, pretty darned close.

I’m also not sure, maybe I screwed up the equations, it seems like there are a lot of unnecessary edge lines in the finished result compared to the example.