Consider that A is a right‐handed inertia coordinate system with unit orthogonal bases i, j and k, and B is a right‐handed body‐fixed coordinate system with unit orthogonal bases i₁, j₁ and k₁. is any vector fixed in A and is a vector fixed in B and equal to prior to the motion of B in A. is a unit vector whose orientation relative to both A and B remains unaltered throughout the motion, and the radian measure θ is the angle when B rotates relative to A. Let and , as shown in Figure AI.1.

The plane π ₁ contains point E and is perpendicular to vector , and point O ₁ is the intersection point of π ₁ and the line including . The plane π ₂ contains point F and is perpendicular to vector , and O ₂ is the intersection point of π ₂ and the line including . Then the point G is in π ₁ and H is in π ₂:

(A.1)

(A.2)

We find a point in plane π ₁ that makes

(A.3)

and

(A.4)

hold, where the three pairwise perpendicular vectors , and form a right‐handed system.

Then

(A.5)

Similarly

(A.6)

Thus, we get

(A.7)

so

(A.8)