Computing Transformation Matrices

The FBX SDK and Maya use the same formula to compute a transformation matrix. However, 3ds Max uses a different formula.

NOTE:The FBX importers and exporters for 3ds Max automatically convert transformation matrices to and from 3ds Max.

FBX and Maya

The following formula represents how the FBX SDK and Maya compute the transformation matrix for a node:

WorldTransform = ParentWorldTransform * T * Roff * Rp * Rpre * R * Rpost ^-1 * Rp^-1 * Soff * Sp * S * Sp^-1

Any given vector is transformed so that V^' = WorldTransform * V.

Where the term:	Is a 4 x 4 matrix that contains:
WorldTransform	Transformation matrix of the node
ParentWorldTransform	Transformation matrix of the parent node
T	Translation
Roff	Rotation offset
Rp	Rotation pivot
Rpre	Pre-rotation
R	Rotation
Rpost ^-1	Inverse of the post-rotation
Rp ^-1	Inverse of the rotation pivot
Soff	Scaling offset
Sp	Scaling pivot
S	Scaling
Sp ^-1	Inverse of the scaling pivot

Notes:

The effect of the formula is that any given vector is first scaled, then rotated, and finally translated.
The R matrix takes into account the rotation order. Because of the mathematical properties of the matrices, R is the result of one of the possible combinations of Rx, Ry, and Rz (each being matrices). For example, for the default rotation order of XYZ, R = Rz * Ry * Rx.

3ds Max

The following formula represents how 3ds Max computes the transformation matrix for a node. All the terms in the formula are the same as in FBX and Maya, except for the three terms that represent geometric transformation:

WorldTransform = ParentWorldTransform * T * R * S * OT * OR * OS

Where the term:	Is a 4 x 4 matrix that contains:
WorldTransform	Transformation matrix of the node
ParentWorldTransform	Transformation matrix of the parent node
T	Translation
R	Rotation
S	Scaling
OT	Geometric transform translation
OR	Geometric transform rotation
OS	Geometric transform scaling

Notes:

Geometric translation, geometric rotation, and geometric scaling relate to the object-offset concept in 3ds Max. These geometric transformations are applied to the node attribute after the node transformations.
Geometric transformations are not inherited: ParentWorldTransform does not contain the OT, OR, and OS of WorldTransform’s parent node.
Geometric transformations are implemented in the FBX SDK as three properties of the FbxNode object: FbxNode::GeometricTranslation, FbxNode::GeometricRotation, and FbxNode::GeometricScaling.