Orbits and locked gimbals (Q354145)

The paper under review is devoted to the popular explanation of parametrizations of the rotation group \(\mathrm{SO}(3)\) and the mechanical aspects of these parametrizations. At first, the author explains two parametrizations of the group \(\mathrm{SO}(3)\) using the Tait-Bryant angles and using the Euler angles. The Tait-Bryant angles are defined as follows. An element of \(\mathrm{SO}(3)\) is decomposed into a rotation about the axis \(x\), then about the axis \(y\) and then about the axis \(z\). The parameters \((t_1,t_2,t_3)\) are the angles of these rotations. Then the mechanical realizations of these parametrizations are discussed. Both these parametrizations can be realized ``by a mechanism with three gimbals or swivel points - one on a fixed axis, the second carried by the first and the third carried by the second''. Using the mechanical realization the orbits of the axes \(x,y,z\) under rotations with parameters \((t,t,t)\) are investigated. This investigation is provided by pictures. The author points out that the orbit of the axis \(y\) has a cusp at \(t=\pi\). The phenomenon of gimbal lock is mentioned. Finally, the global topology of the group \(\mathrm{SO}(3)\) is discussed.