Fig 1 is a classic crochet hook rim, with a midrange fit. Here you can easily understand if this tire was not concentric, the bead would pop out of the rim tire well. Blowing off the rim.
Fig 2 shows a rim at a maximum diameter, where the bead is firmly seated at the sides of the tire well.
Fig 3 is the minimum rim diameter condition with the bead at the hook. All three of these conditions are possible with rims lacking a bead shelf. The purpose of a bead shelf is to more readily center the bead on the rim. This eliminates the need for a hook as the tire centers automatically as the sidewalls move outward from tire pressure.
This is an illustration showing what we see as the basic flaw in hooks, essentially hooks are trying to constrain the bead externally. This is like dropping a ring into a conical bowl, it is very unlikely that the bead will find a common central axis with the cone of the bowl.
This illustration clearly demonstrates a ring on a cone finds a common central axis, this is what a bead shelf does, and this is why motorcycles and automobiles adopted this bead well form 50 years ago.
The image here shows a typical aircraft wheel, this is both safety critical and very high pressure, >200psi. No hooks needed and it’s safe light and air tight.
This image is the original Geax Speedlock rim, this was the first ever bead lock, bead shelf rim. Developed by Josh Deetz and patented by Vittoria. At this time the trust was to try to move users to explicit bead seat, and wide hook width rims to best support the connection between rim and tire. This rim allowed one-handed mounting and demounting of tires, even tight beaded foldable tires. The perfect rim profile should allow very easy mounting and consistent self-centering during inflation. Hookless versions of this rim were CNC machined to determine the need for hooks in the presence of a bead shelf and hump. A decade later hookless rim development started with Paul Lew.
A more modern high-performance hookless rim trend started with Paul Lew from Lew racing. He adopted a automotive like bead shelf, and ushered in a new design standard for carbon rims. Some original outlines of these developments can be downloaded here:
LEW Clincher White Paper Final 1.0 & Chart 1-8