I became interested in the math of curved Tunisian crochet objects while making the above cat bed. (It has a flat hollow in the center, so my elderly, always-cold cat can nestle into the center of it.)
I created the curvature by crocheting seven (7) spiraling sections with increases between them
and by varying the height-to-width ratio of the stitch combo.
I used a shorter stitch combo for the central flat section
and a taller stitch combo for the part that curved to form a hollow.
(Actually, if I had put more stuffing inside, this section would have curved to form a bump.)
I also used rows of stitches with no increases to create flat sides
and 7 sections with decreases instead of increases to form a flat bottom.
I expected that it would be easy to figure out how many sections I needed to get a flat round using simple assumptions. Hah!
I expected that Pi = 3.1416 increases per round would create a flat object for stitches that are roughly as wide as they are high (like the Simple Stitch), because the circumference-to-diameter ratio is what defines Pi.
And that I would need C increases per round to make the work flat for an arbitrary stitch combo,
where C = Pi(h/w) and h/w is the height-to-width ratio of the stitch combo.
Unfortunately, when I compared this theory with my experience and measurements taken from pictures of the cat bed…
So, it looks like you need something like C = 4.5(h/w) sections
to make the object flat,
and more sections will create a frilly object
and fewer sections will mean the object has a cupped shape.
…And I could obsess over why C appears to equal 4.5 rather than 3.1416,
but I choose not to.