spatial rotation R as, Then the composition of the rotation RB with RA is the rotation RC=RBRA, with rotation axis and angle defined by the product of the quaternions, that is Expand this quaternion product to Divide both sides of this equation by the identity resulting from the previous one, and evaluate This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two component rotations. He derived this formula in 1840 (see page 408). The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles. Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and angle by Its components can be expressed in terms of the components of a unit quaternion representing the same rotation as The modified Rodrigues vector is a stereographic projection mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane. The projection of the opposite quaternion -q results in a different modified Rodrigues vector than the projection of the original quaternion q. Comparing components one obtains that Notably, if one of these vectors lies inside the unit 3-sphere, the other will lie outside. Cayley–Klein parameters See definition at Wolfram Mathworld. Higher-dimensional analogues Vector transformation law Active rotations of a 3D vector p in Euclidean space around an axis n over an angle η can be easily written in terms of dot and cross products as follows: wherein is the longitudinal component of p along n, given by the dot product, is the transverse component of p with respect to n, and is the cross product, of p with n. The above formula shows that the longitudinal component of p remains unchanged, whereas the transverse portion of p is rotated in the plane perpendicular to n. This plane is spanned by the transverse portion of p itself and a direction perpendicular to both p and n. The rotation is directly identifiable in the equation as a 2D rotation over an angle η. Passive rotations can be described by the same formula, but with an inverse sign of either η or n. Conversion formulae between formalisms Rotation matrix ↔ Euler angles The Euler angles can be extracted from the rotation matrix by inspecting the rotation matrix in analytical form. Rotation matrix → Euler angles