Different transform matrices in blender: Object matrix_basis/matrix_local/matrix_world/matrix_parent_inverse; Bone matrix/matrix_local; PoseBone matrix/matrix_local/matrix_channel;

transform_matrix_in_blender.txt

# Different transform matrices in blender

Object matrix_basis/matrix_local/matrix_world/matrix_parent_inverse; Bone matrix/matrix_local; PoseBone matrix/matrix_local/matrix_channel;

## 1. Object matrix

      """
    Refer to:
    https://docs.blender.org/api/current/bpy.types.Object.html?highlight=object#bpy.types.Object
    """

matrix_basis
 (从object space 到 world space的变换矩阵，计算的是未关联parent时的坐标，关联parent后一直报错关联前的值不变）
Matrix access to location, rotation and scale (including deltas), before constraints and parenting are applied
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0))

matrix_local  （=matrix_parent_inverse*matrix_basis）
(从object space 到 parent object space的变换矩阵，未关联parent时一直等于matrix_basis（相当于parent就是world space 的原点）
Parent relative transformation matrix. Warning: Only takes into account object parenting, so e.g. in case of bone parenting you get a matrix relative to the Armature object, not to the actual parent bone
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0))

matrix_parent_inverse 
（关联parent时parent.matrix_world的逆矩阵，不是parent.matrix_world的逆矩阵，因为parent可能还有自己的parent，
为了方便在matrix_basis发生变化时在线更新matrix_local=matrix_parent_inverse*matrix_basis）
Inverse of object’s parent matrix at time of parenting
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((1.0, 0.0, 0.0, 0.0), (0.0, 1.0, 0.0, 0.0), (0.0, 0.0, 1.0, 0.0), (0.0, 0.0, 0.0, 1.0))

matrix_world  （=parent.matrix_world * matrix_local）

(从object space 到 world space的变换矩阵，未关联parent时一直等于matrix_basis（相当于parent就是world space 的原点）
Worldspace transformation matrix
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((1.0, 0.0, 0.0, 0.0), (0.0, 1.0, 0.0, 0.0), (0.0, 0.0, 1.0, 0.0), (0.0, 0.0, 0.0, 1.0))

https://blender.stackexchange.com/questions/131135/matrix-basis-vs-matrix-local-for-object-parented-to-bone

* matrix_basis is simply a way to access the pose-bone or objects (location,scale,rotation) properties.
This is just for convenience - so you don't have to consider if euler/quaternion/axis-angle rotation are used when getting/setting a transformation.

* matrix_local is the the objects world-space 4x4 matrix, relative to it's parent, if it has a parent - if the parent is a bone or a vertex, it will be relative to the parent matrix defined by the vertex/bone parent relationship.


https://blender.stackexchange.com/questions/35125/what-is-matrix-basis

Object.matrix_local = Object.matrix_parent_inverse * Object.matrix_basis 
Object.matrix_world = Object.parent.matrix_world * Object.matrix_local

如果没有父节点： Object.matrix_parent_inverse = Identity_4x4
Object.matrix_local =Object. matrix_basis = Object.matrix_world

对于object上的一个顶点（其在objcet space的坐标为: pos_objspace)，其在local space （即parent space）的坐标（如果没有parent，那local space就是world space)：

pos_localspace = Object.matrix_local * pos_objspace = Object.matrix_parent_inverse * Object.matrix_basis * pos_objspace 

Object.matrix_basis * pos_objspace: 得到的是在绕object自身的轴旋转之后的世界坐标，因为matrix_basis最后一列包含了object当前的世界坐标；
Object.matrix_parent_inverse * Object.matrix_basis * pos_objspace： 将旋转之后的世界坐标映射到local space (parent space)，parent object自身的局部坐标系

pos_worldspace = Object.matrix_world * pos_objspace = Object.parent.matrix_world * Object.matrix_local * pos_objspace
Object.matrix_local * pos_objspace： 这一步将object space 旋转平移变换之后的坐标变换到local space (parent space)
Object.parent.matrix_world * Object.matrix_local * pos_objspace: 这一步将local space  (parent space)旋转平移变换之后的坐标变换到world space


https://blender.stackexchange.com/questions/180527/how-to-calculate-a-parent-inverse-matrix
Object.matrix_parent_inverse = Object.parent.matrix_local.inverted (wrong, 经过代码验证，这里在2.8里是不成立的)

Object.matrix_parent_inverse = Object.parent.matrix_world.inverted (right, 经过代码验证，这里在2.8里是成立的)


## 2. Bone matrix (Rest Bone)
rest_bones: armature_object.data.bones (bpy.types.Bone)
 	refer to: 
    https://docs.blender.org/api/current/bpy.types.Armature.html?highlight=armature#bpy.types.Armature.bones
    https://docs.blender.org/api/current/bpy.types.ArmatureBones.html#bpy.types.ArmatureBones
    https://docs.blender.org/api/current/bpy.types.Bone.html#bpy.types.Bone

matrix
（bone space 跟parent bone space之间的变换矩阵）
3x3 bone matrix
Type
float multi-dimensional array of 3 * 3 items in [-inf, inf], default ((0.0, 0.0, 0.0), (0.0, 0.0, 0.0), (0.0, 0.0, 0.0)), (readonly)

matrix_local
（从bone space到armature object space的变换矩阵）
4x4 bone matrix relative to armature
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0)), (readonly)


## 3. PoseBone matrix
pose_bones: armature_object.pose.bones (bpy.types.PoseBone)
        refer to: 
        https://docs.blender.org/api/current/bpy.types.Object.html?highlight=types%20object#bpy.types.Object.pose
        https://docs.blender.org/api/current/bpy.types.Pose.html?highlight=pose#bpy.types.Pose
        https://docs.blender.org/api/current/bpy.types.PoseBone.html?highlight=posebone#bpy.types.PoseBone

bone
Bone associated with this PoseBone
Type
Bone, (readonly, never None)

matrix （PoseBone.matrix = PoseBone.parent.matrix * PoseBone.parent.bone.matrix_local.inverted() * PoseBone.bone.matrix_local * PoseBone.matrix_basis）
（从bone space到armature object space的变换矩阵）
Final 4x4 matrix after constraints and drivers are applied (object space = armature space)
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0))

matrix_basis
（在bone space发生旋转/位移的变换矩阵，相对于rest bone）
Alternative access to location/scale/rotation relative to the parent and own rest bone
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0))

matrix_channel （PoseBone.matrix_channel = PoseBone.matrix * PoseBone.bone.matrix_local.inverted()）
（为了方便计算 armature空间任意一点受该骨骼旋转之后的坐标在armature空间）
4x4 matrix, before constraints
Type
float multi-dimensional array of 4 * 4 items in [-inf, inf], default ((0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0)), (readonly)

经过代码验证，这里在2.8里是成立的:
PoseBone.matrix_channel = PoseBone.matrix * PoseBone.bone.matrix_local.inverted()
PoseBone.matrix = PoseBone.matrix_channel * PoseBone.bone.matrix_local

PoseBone.matrix = PoseBone.parent.matrix * PoseBone.parent.bone.matrix_local.inverted() * PoseBone.bone.matrix_local * PoseBone.matrix_basis

*_armature: vector in armature space
*_bone: vector in bone space

V_armature_after_rot = PoseBone.matrix * V_bone
			   = PoseBone.matrix_channel * (PoseBone.bone.matrix_local *  V_bone)
		           = PoseBone.matrix_channel *  V_armature 

上式可以方便计算 armature空间任意一点受该骨骼旋转之后的坐标在armature空间，

比如用骨骼驱动mesh，先把mesh上的点映射到armature空间，再将新的坐标左乘以该矩阵，得到该点在armature空间新的位置，再左乘armature 的matrix_world即得到变换之后的全局坐标，
整个过程如下：

V_world_after_rot = ArmatureObject.matrix_world * ( ArmatureObject.pose_bones[ii] * ArmatureObject.matrix_world.inverted() * MeshObject.vertex.matarix_world *  MeshObject.vertex[jj].loc)
V_mesh_after_rot = MeshObject.vertex.matarix_world.inverted() * V_world_after_rot
 = MeshObject.vertex.matarix_world.inverted() * ArmatureObject.matrix_world * ( ArmatureObject.pose_bones[ii] * ArmatureObject.matrix_world.inverted() * MeshObject.vertex.matarix_world *  MeshObject.vertex[jj].loc)

定义M_mesh2arm = ArmatureObject.matrix_world.inverted() * MeshObject.vertex.matarix_world, 该矩阵将mesh空间的点映射到armature空间
M_arm2mesh= M_mesh2arm.inverted() = MeshObject.vertex.matarix_world.inverted() * ArmatureObject.matrix_world , 该矩阵将armature空间的点映射到mesh空间

则上述公式可以写作：
V_mesh_after_rot = M_mesh2arm.inverted() * ( ArmatureObject.pose_bones[ii] * M_mesh2arm *  MeshObject.vertex[jj].loc)

https://stackoverflow.com/questions/1273588/exporting-keyframes-in-blender-python

The channel data should be applied on top of the bind pose matrix.
The complete formula is the following:
Mr = Ms * B0*P0 * B1*P1 ... Bn*Pn
where:
Mr = result matrix for a bone 'n'
Ms = skeleton->world matrix
Bi = bind pose matrix for bone 'i'
Pi = pose actual matrix constructed from stored channels (that you are exporting)
'n-1' is a parent bone for 'n', 'n-2' is parent of 'n-1', ... , '0' is a parent of '1'

## 4. EditBone Matrix
edit_bones: armature_object.data.edit_bones (bpy.types.EditBone)

    """ 
    refer to: 
        https://docs.blender.org/api/current/bpy.types.Armature.html?highlight=armature#bpy.types.Armature.edit_bones
        https://docs.blender.org/api/current/bpy.types.ArmatureEditBones.html#bpy.types.ArmatureEditBones
        https://docs.blender.org/api/current/bpy.types.EditBone.html?highlight=edit_bones

    """