than the matrix representation and less susceptible to round-off errors The quaternion elements vary continuously over the unit sphere in , (denoted by ) as the orientation changes, avoiding discontinuous jumps (inherent to three-dimensional parameterizations) Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric functions It is simple to combine two individual rotations represented as quaternions using a quaternion product Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a matrix. Quaternions also capture the spinorial character of rotations in three dimensions. For a three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a rotation changes from a scalar +1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two full turns). This cycle repeats every 2 turns. After turns (integer ), without any intermediate untangling attempts, the strings/bands can be partially untangled back to the turns state with each application of the same procedure used in untangling from 2 turns to 0 turns. Applying the same procedure times will take a -tangled object back to the untangled or 0 turn state. The untangling process also removes any rotation-generated twisting about the strings/bands themselves. Simple 3D mechanical models can be used to demonstrate these facts. Rodrigues vector The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) can be expressed in terms of the axis and angle of the rotation as follows: This representation is a higher-dimensional analog of the gnomonic projection, mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane. It has a discontinuity at 180° ( radians): as any rotation vector tends to an angle of radians, its tangent tends to infinity. A rotation followed by a rotation in the Rodrigues representation has the simple rotation composition form Today, the most straightforward way to prove this formula is in the (faithful) doublet representation, where , etc. The combinatoric features of the Pauli matrix derivation just mentioned are also identical to the equivalent quaternion derivation below. Construct a quaternion associated with a