How does blender rotate parented bones?

Question

I am trying to understand how exactly parented bones are rotated in blender.

Consider the following hierarchy: Bone1 -> Bone2 -> Bone3 -> Bone4 -> Bone5

If i were to rotate Bone2, how will blender rotate it & it's children bones(Bone3, Bone4, Bone5)? I'd assume that Blender does the following:

1. Set pivot point of Bone2 to the bones local position, Rotate Bone2
2. Set pivot point of Bone3 to (Bone2.localPos + Bone3.localPos) Rotate Bone3
3. Set pivot point of Bone4 to (Bone2.localPos + Bone3.localPos + Bone4.localPos) Rotate bone4
4. Set pivot point of Bone5 to (Bone2.localPos + Bone3.localPos + Bone4.localPos + Bone5.localPos) rotate bone5

And from what i understand the local position of a bone is calculated the following way:

Bone.localPos = Bone.pos - Bone.parent.pos

If the bone has no parent then it's just the bones position

An example with actual position values:

Now even more explicitly:

Assume that the bones have the following global positions:

Bone1: (50, 20, -25)
Bone2: (35, -15, 20)
Bone3: (-10, -20, 50)
Bone4: (-35, 20, 40)
Bone5: (15, 25, 10)

The local positions would be:

Bone1Local = (50, 20, -25)
Bone2Local = (-15, -35, 45)
Bone3Local = (-45, -5, 30)
Bone4Local = (-25, 40, -10)
Bone5Local = (50, 5, -30)

The pivot points when Bone2 is rotated would be the following:

Bone2: (-15, -35, 45) // bone2Local
Bone3: (-60, -40, 75) // bone2Local + bone3Local
Bone4: (-85, 0, 65) // bone2Local + bone3Local + bone4Local
Bone5: (-35, 5, 35) // bone2Local + bone3Local + bone4Local + bone5Local

If this is not how blender rotates bones, an explanation how it actually works would be much appreciated, if it is please just confirm that my understanding of it is correct :)

Blend file:

Answer 1

We can make the exact armature you'd like to use as an example. It looks like this:

However, you didn't specify in what axis you wanted to rotate bone2. That would let you actually do some math and compare with what you see.

The pivot point for all bones, in your example, where we rotate bone 2, is world space 35,-15,20. That is the location of the head of bone 2. Anything that happens, to any bone in this unconstrained armature, as a result of the rotation of bone 2, will be a rotation about that point. We can see this. Let's duplicate the armature:

On the left, I have rotated your Bone2 -45 degrees in my view axis. On the right, I have the same bone locations, but all bones are unparented (see outliner, and forgive me for not fixing names); in this view, I snapped the cursor to the location of your Bone2, then rotated it and all bones that would have been parented to it, by -45 degrees in the view axis, about the cursor.

If we want to convert the location of Bone2's head from a world space position to a position local to one of those bones, we can (it depends on the axes of the bones; you're making some unspecified assumptions about the axes.) But there is no reason to do that, and it would be misleading. The origin of the inherited rotation is not stored as some local space number. While those armatures that I showed have the same final rotation, they wouldn't interpolate from rest to that position in the same way. No object or bone contains a history of rotations and origins of those rotations; they only ever contain a single rotation, about their own (untransformed) origin. The transformation of the parent is inherited by the children only via the parenting relationship. The child does not ever know where the parent's origin is.

If for some reason, I feel the need to know where the origin of this particular inherited rotation is, that is to say, I need to know the local space position of the head of Bone2, it's possible. I can give a bone a copy position constraint, world->world, targeting Bone2, then apply as visual transform. The local space position will be the transform the bone gets:

Here, we can see on the sidebar the Bone3 local space position of Bone2. As I said, this value depends on the axes of Bone3.

If you want to know more about how parenting works, you first have to understand how objects are transformed. I would recommend starting at https://en.wikipedia.org/wiki/Transformation_matrix . The parent has a matrix representing its transform. The child inherits the parent's transform by multiplying its own transform with that of its parent. If the child has not been transformed, then its transform is its parent's transform, the exact same transform, but in matrix form, not keyframe triplets.

I would also strongly recommend https://gamemath.com/book/intro.html where you can find a professional primer on 3D math (including understanding 3D transformations) that the authors have chosen to share for free online. I have a physical copy and have found it invaluable. That's longer than Wikipedia, but that's a good thing, not a bad thing.

Answer 2

If your bones doesn't have any constraints or locked axis, if the parent bone will rotate (in pose mode) around its head, every children will move with it. So it works exactly the same as if you would parent to objects to each other. Just the objects take their pivot point, and bones will take their head as rotation point.

Same is for movement, if the bone can be moved.

In your case: the rootjoint is parent of all, so if you rotate it, your whole model + armature rotates arount its head.