Looking at google pics it seems like they very a little bit if they come to a point or not so that is going to be hard to find a cad profile for the requirements. I did find a reference here though with a description, I used a point in this example.

https://emeralds.com/education/all-about-emerald-cabochons/

Bemfarmer already showed the source..

Pilou, bemfamer, Death, Burrman, Phiro, Grendel, you are awesome and your input
is very much appreciated.

This Cabochon shape is really interesting, it looks like four symmetrical planes with
a curved tip and the very top unifying all four surfaces being additionally rounded in
a bit different way.

In regard of this endeavour I stumbled over the great little Squircle Script - is there a
difference in it to the "perfect ellipse" script which I was unable to find ?

TOM

I think that the "super-ellipse" has more generalized, elaborate mathematics, than the squircle script.
(The squircle is a subset of the super-ellipse...)
(I like Burr's method, for the Cab.)

The squircle shape for the outline of the gem is nice.
The lower portions of the Cab gem seem to vary, and are important for mounting...
Read that the bottom is often not polished, for light reasons, and can be flat or rounded.
It is difficult to cut, for perfect symmetry. Have not seen the special tool machine which could be used...

- Brian

The Squircle (still strugelling with this word ;-)) script is really useful.

Taking a closer look at available photo material it seems
that the 4 edges start almost completely rounded at
the bottom sharpening towards the tip. The tip itself
being a surface of its own filling the gap between
the four surfaces.

Hi Tom,
perhaps this method will give you the rounded peak needed.
I started with a square and added mid points to each side.
Then selected the mid points and scaled to give the desired curvature,
Offset copies made and added point for peak then lofted as per Burr's video.
Added a fillet running diagonal from corner to corner.
The other diagonal same radii but requires a second fillet set for the peak area to be slightly smaller.
hope that makes sense.
Cheers
Barry

Here is another option...

Since it seems the surgarloaf is more of an "artistic grind" and not really a mathematically produced result, you can change the base shape and all the angles of the surface structure, along with Blend parameters to "Eyeball" some various results... All smooth.

Sugarloaf cut video. Not in English (India).
https://www.youtube.com/watch?v=dsfqdqgxEyE

- Brian

Google search of "Cabo Gem Cut" shows several videos, (but not for sugarloaf), and the grinding equipment.

Amazing approach, Burrman. The video shows you using the Superellipse script.
Is there any difference to the Squircle script ?

It is Burr's video that you refer to?

- Brian

Uh I guess I mixed up your names ... Yes, the Video shows Burrman using
the Superellipse script and as I did not find this one (I am using the "Squircle")
I wondered about the differnces.

The superellipse2d script is from 2013, and includes a squircle, I think as a subset.

https://moi3d.com/forum/index.php?webtag=MOI&msg=5616.1

About February 2022, a different squircle, using a different math formula, was created, the Fernandez-Guasti squircle. (Later in above link.)
This version is not a subset of superellipse2d. (?)

- Brian

I looked up the pronunciation of squircle. Combine the rodent squirrel with ircle.
(still hasn't made it into spellchecker :-)

On the first picture of this thread and in the video we see, the Sugarloaf Cabochon have 4 faces with edges.
With the video, I discover the render of this cut, that I see very rarely in stores.
Very nice cut...

Thank you Disovered lots if scripts at the link you posted and will give them a try.

Hi Tom,
Brian is the creator of these scripts. He has made several of the "mathematical formula" scripts that create well known geometrical shapes.

He is best to answer about them and if there is any difference (or similarity)

Since this impossible word "Squircle" nauseated my mind it mixes up the names Burrmen Benfarmer and Brian -sorry ;-)

Hey Tom... makes it harder to have to translate everything. Hang tough!

"Squircle"

Square + circle

Squ ircle

... another elegant way

Mik

This link shows a side view of a cabochon:
https://www.rockngem.com/filling-an-empty-space-and-enhancing-a-cabochon/
I had thought that it is a sugarloaf, but now think it is a "slab" for a pendant. ?

Learned a new word: "vug"
https://en.wikipedia.org/wiki/Vug



- Brian

Very nice Mik

Due to your kind collaboration this thread turned into a nice
compilation of how to achieve one goal by different means -
thank you all !!

As one contributor wrote, the shape of a "squircle" is aesthetically
very appealing. I wanted to apply this theme of "squaring the circle"
to the whole shape, starting with a circle at the bottom morphing
into a slightly rounded square at the top. The how to is attached
below.