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While watching this clip of Harry wearing the Sorting Hat, I understood that the Sorting Hat has a lot of fidelity in the animation of its mimic and can even gesture.

With Google, I did not find any information on its construction.

Is there any information or analysis which says how many joints/bones are needed for that?

Obviously it needs much more than these cheap 5-servo Sorting Hats have.

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The sorting hat was largely CGI. There seems to have also been a puppet hat, but it was operated as a hand-puppet (with the operator simply obstructed by the person wearing it or digitally painted out) and as far as I can tell, there was never an animatronic hat.

But the Hogwarts Sorting Hat was mainly a creation of CG crews. "We gave the animator first crack at designing it," says [Production designer Stuart] Craig. "It's really quite ingenious, the way that the eyes and mouth are just parts of its folds."

Entertainment Weekly: 23 Astounding Harry Potter Slides

We also have a post from Reddit User noramacsbicth (AKA Nora Koonce, daughter of Russell Koonce, Digital Effects Artist for the various potter films) in which she described the hat thusly;

I got to wear the ACTUAL sorting hat. Depressingly enough, it was just a plain brown hat shaped vaguely like the sorting hat- most of it was done completely with visual effects.

You can see an earlier version of the hand-puppet here. This one never made it into the film

enter image description here

As to the joints on the CGI model, it's probably instructive to look at the CGI puppet for Dobby (animated by the same studio that animated the hat). As you can see there are thousands of potential points of articulation rather than a singular skeleton inside the model. Each point can be deformed by the animators to give the effect of movement. There's no good reason to assume that they used a different technique for the hat.

enter image description here

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  • I am aware that most was done with CGI, but also this had to be animated (in software).
    – Torsten Knodt
    Sep 20, 2020 at 14:59
  • Apologies for the non-English translation on the making-of. I can't seem to find and English-speaking version.
    – Valorum
    Sep 20, 2020 at 18:02
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    @TorstenKnodt not all computer animation is skeletal animation. In 2002 the most prominent facial animation tech would be morph target animation (also used to animate Gollum's face in LotR movies, in conjunction with motion capture). This technique does not require a skeleton structure, and Dobby model shown in linked video seems to use the same approach.
    – Danila Smirnov
    Sep 21, 2020 at 7:07
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    @DanilaSmirnov - The sorting hat in later films explicitly didn't use motion capture. It was hand-animated using 'keyframe technology'. I'm not sure about the earlier films though, so I've excluded that from the answer.
    – Valorum
    Sep 21, 2020 at 7:30

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