Classic techniques have been established to characterize N × N proper orthogonal matrices using the N -dimensional Euler’s theorem and the Cayley transform. These techniques provide separate descriptions of N -dimensional orientation in terms of the constituent principal rotations or a minimum-parameter representation. The two descriptions can be linked by the canonical form of the extended Rodrigues parameters. This form is developed into a new minimum-parameter representation that directly links to the principal rotations. The new representation is solved using analytic and geometric approaches for N = 3 and N = 4 , and numerical solutions are found for N= 5. In fact multiple solutions, which are related geometrically by different coordinatizations of the principal planes, have been found. The new parameters represent a projection of the principal rotations onto the planes formed by the body coordinates.

中文翻译：

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N 维主旋转的最小参数表示

已经建立了使用 N 维Euler定理和Cayley变换来表征 N × N个 固有正交矩阵的 经典技术 。这些技术根据组成主旋转或最小参数表示来提供对 N 维方向的单独描述 。可以通过扩展的Rodrigues参数的规范形式来链接这两个描述。该形式被开发为直接链接到主旋转的新的最小参数表示形式。对 N = 3和 N = 4 使用解析和几何方法求解新的表示 ，并找到了数值解。 N ＝ 5。实际上，已经找到了多个解，这些解在几何上是通过主平面的不同配比而相关的。新参数表示主旋转在由身体坐标形成的平面上的投影。