We are considering the rotation that goes from a coordinate system A to a coordinate system B, noted BfromA. This notation allows to combine transforms without ambiguity, e.g BfromC = BfromA * AfromC, and applies naturally to a point on the right: pointInB = BfromA * pointInA

Here they are defined in the Z-Y-X order, Tait-Bryan, Intrinsics. Note that this is the same as x-y-z, Tait-Bryan, Extrinsics.

BfromA = R(Z) * R(Y) * R(X)

This corresponds to the rotation of an object A in a coordinate system B. Applying the extrinsics sequence this means that a point gets transformed by first applying the rotation around XIf you are trying to simulate the transform with your fingers, applying these angles in sequence will make your finger-axes go from B to A, not from A to B.

X = | |

Y = | |

Z = | |

The rotation is uniquely defined (up to the sign) by a unit axis [X,Y,Z] and an angle Θ around it.

X = | |

Y = | |

Z = | |

Θ = | |

A unit quaternion is a unique (up to the sign) representation. Quaternions allow easy interpolation and composition. See the wikipedia page.

X = | |

Y = | |

Z = | |

W = | |

Rotation matrices are 3x3. The **row vectors** are the coordinates of each axis of B expressed in A. The **columns vectors** are the coordinates of each axis of A expressed in B.

$$[X,Y,Z]_B = [Y, -Z, X]_A$$