I only made curved walls once since becoming a basement specialist in 2004. Most of these photos are from that project. We were still using a polystyrene ICF, in those days.
With hindsight, it would have been more work to reduce the width of the upright boards inside to get the radius required but much easier because we could have simply screwed the horizontal straps to timber.
Before that, on big projects, I saw them formed a few times.
The top had to be corbelled out so that, starting inside, he could have a bearing for a beam and block floor; then the waterproof concrete upstand upon which inner skin blockwork was to be built, then space for insulation that would continue into his cavity wall, finally the extra corbel is for face brickwork.
Clearly this ICF. But we could not just rely on the plastic ties, we had to seriously brace with timber - which is why it did not burst or go out of shape.
Any radius is achievable. Radius curves, parabolas or S shapes.
But the theory of what goes on is entirely different to a straight wall.
In a straight wall the threaded rods control the width of the concrete and the props merely have to keep the formwork upright for all the weight to go down to the solid floor beneath. I always say it is like holding a scaffold pole upright. As long as it does not begin to lean over it is easy to keep it where it is; but if it leans more than a few inches it is very heavy indeed and may defeat you and fall over.
A curve is like a straight already leaning over. The weight of the concrete will find that it can spread and increase the circumference length. Because the boards need to be vertical the weight of the concrete will try to push the boards out, making the wall lean over thus getting a little more concrete in. And if that process was allowed to begin it would burst the wall open in a moment.
I do not fill walls with concrete in one go because I could neither properly deliver nor properly compact concrete 3m down through 2 lines of reinforcing steel. For all sorts of failures of workmanship the concrete would not be waterproof.
An additional set of difficulties with the ICF I used was accommodating a precast concrete floor inside as well as waterproofing horizontally with an upstand.
We began with full boards that we cut off on the outside to the underside of the corbel.
When we filled with concrete the first time we only filled to the underside of the reinforcing steel yet to be tied on to support the corbel. Therefore there was not going to be much concrete pressure at the top which is why the strapping got further apart. The bracing off the excavation was necessary because it was difficult to fix the end of the curve on the outside strongly to the end of the straight that meets it. The curve could have survived but moved as a single entity.
And so to the process in chronological order.
Curved walls will be far more time consuming than straight walls. But in terms of materials the cost will be about the same. A small bit of curved wall might add panache to your project worth the extra time it took to form.