Parameterizations
The one-to-one correspondence between proper and improper rotations (based on the fact that in three dimensional space each improper rotation is a composition of the inversion and a unique proper rotation) allows us to limit the consideration of parameter
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HE one-to-one correspondence between proper and im proper rotations (based on the fact that in three dimensional space each improper rotation is a composition of the inversion and a unique proper rotation) allows us to limit the consideration of parameterizations to proper rotations. Using an additional discrete parameter pointing out whether the transformation involves the change of handedness or not, the improper rotations can be easily taken into account. Proper rotations can be parameterized by a 3 x 3 special orthogonal matrices. However, this parameterisation is highly redundant. As we mentioned earlier, because of the orthogonality conditions, the number of independent parameters is 3, and three parameters suffice to determine a rotation. Is it possible to have aglobaI "nice" (one-to-one, continuous, with continuous inverse) parameterization which would map the rotation group into the 3 dimensional Euclidean space? The answer is no, and this follows from topological arguments (Stuelpnagel, 1964). A number of most frequently used parameterizations will be described in some detail. Many of them are constructed in such a way that singularities show up through multiple parameter sets corresponding to certain rotations. In most cases these rotations will be half-turns. As will be seen later, halfturns are in a sense special among rotations.
2.1 Half-turns It is easy to see that if an orthogonal matrix 0 is symmetrie, then the corresponding rotation is a half-turn. Apply 0 (=f. I) to both sides of the symmetry condition 0 = OT. This gives 00 = I. Thus, 00 corresponds to the complete-turn and 0 corresponds to a half-turn. The opposite statement is also true: if a rotation is a half-turn, it is represented by a symmetrie special orthogonal matrix. Since a half-turn 0 applied twice gives the complete-turn wh ich is equivalent to null rotation, the halfturn is identical with its inverse (0 = 0- 1 ). By its orthogonality, the matrix must be symmetrie: 0 = OT.
A. Morawiec, Orientations and Rotations © Springer-Verlag Berlin Heidelberg 2004
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2 Parameterizations
Moreover, the trace of a symmetric special orthogonal matrix is equal either to 3 or to -1. This can be easily demonstrated: Since the matrix is symmetric, its eigenvalues are real. Real eigenvalues of a special orthogonal matrix are equal to ±1, at least one of them is equal to 1, and their product equals 1. Hence, if all eigenvalues are equal to 1, the trace is 3. If one eigenvalue equals 1 and the other two are -1, the trace is -1. Obviusly, only the identity matrix has the trace of 3. It will be evident later that the half-turns are the only proper rotations with the corresponding matrices having the trace of -1.
2.2 Cayley Transformation and Rodrigues Parameters Let x' = Ox, where x is a vector and 0 is a special orthogonal matrix. The invariance of the vector's magnitude x' . x' = x . x can be written as the orthogonality condition y . z = 0 of vectors y and z defined by y = x - x' = (I -O)x and z = x+x' = (I +O)x. For non-singular 1+0, x can be
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