The technique is called 'kerf bending' because multiple kerf-width cuts are made to allow the wood to bend. If you google that you will find lots of good examples. Another term that is used is 'living hinge'. Here's a nice blog post on the topic. https://www.martin-breuer.com/kerf-bending-patterns
As far as designing it goes, the easiest option is to try different patterns of cuts and see what works best. If you look around online you'll get some ideas. It's difficult to say what pattern and how many cuts are needed for a particular curve since every material is different and it would require some higher level knowledge to actually calculate the stresses and strains, etc, therefore, my recommendation is to try it and see what works.